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Citation preview
Beware of the Sucker Answer
Make sure you answer the question that is asked
Double check the question before you fill in the bubble
Do the Easy Ones First Then go Back and do the Hard Ones
For Multiple Choice TestsbullCheck each answer ndash if impossible or silly cross it out
bullBack plug (substitute) ndash one of them has to be the answer
bullFor factoring ndash Work the problem backwards
bullSketch a picture
bullGraph the points
bullUse the y= function on calculator to match graphs
Read the directions
for the test carefully
Read each question carefully
FAQs
Number next to a Letter (variable) means Multiply
4x
If x = -3
Then substitute amp multiply
4(-3) = -12
12a + -2b
If a = 5 b = -3
Then substitute amp multiply
12(5) + -2(-3) = 66
So do numbers next to ( ) and letters next to letters xy = xy
4(x) = 4x
(-2)(a) = -2aab = ab
4f g = 4fg
Addition and Subtraction are snobs They just combine with their own kind They form cliques
3x ndash 3a + 7 + 6x + 2a = 9x ndash a + 7
4xsup2 + 6x +9 ndash 15x + 3xsup2 + 10 = 7xsup2 - 9x + 19
Thatrsquos Just How They Do Thatrsquos How It Is
Deal With It
Multiplication and Division are party animals They will do it with anyone2x 4y = 8xy a bc = abc 12xy4x = 3y
Understanding Algebraic Culture is the Key to Success
23 = 2 divide 3
Numbers come with Signs (+ -)The sign is in front
Remember These are the same thing
5 ndash 3 = 25 + - 3 = 2
Because Subtraction is Adding the Opposite
If you are confused Circle the number amp the sign in front then do the math
2 ndash 56 + 7 ndash 8 ndash 10 = - 65
ABSOLUTE VALUE
The absolute value is always positive
The absolute value of 5 is 5The absolute value of -5 is 5
To solve drop the bars and make the inside number positive
Watch Out -6 = - 6 because the negative is lurking outside the bars
ALGEBRA OPPOSITES
Opposite of Multiplication is Division
25 4 = 100 100 divide 4 = 25
Positive Negative Numbers
The opposite of -5 is 5
The opposite of 6 is -6
Opposites add to zero
-4 + 4 = 0
Opposite of Addition is
Subtraction12 + 18 = 30
30 ndash 18 = 12
Opposite of Squaring is Square Rooting = 25 = 5
Adding Negative numbers - Think MONEY $$$$$$ You wonrsquot miss it
Algebra Truths
UGLY numbers work the
same as PRETTY Numbers
You canrsquot add FROGS amp
SMILEY FACES
LETTERS amp NUMBERS
work the same
Simplifying is cleaning up
your room put all the
FROGS together amp all the
SMILEY FACES together
then do the Math
No Equal Sign Simplify It Combine Like TermsSimplifying is like cleaning up your room put all the FROGS (Variables) together and all the SMILEY FACES (Numbers) together then do the Math Donrsquot Forget Numbers come with Signs
4x -5 +6x +21 -8x
4x-5
+6x+21
-8x
--------------------------------------------------------------------------------------------------------------------------------4x +6x -8x -5 +21
4x +6x -8x-5 +21
-------------------------------------------------------------------------------------------------------------------------------- 2x +16
2x +16
You canrsquot add FROGS amp SMILEY FACES so you are finished Good Job
WATCH YOUR SIGNS
Adding or Subtracting
If signs are the same add them and use the sign45 + 34 = 79 - 45 ndash 10 = -55
If signs are different subtract and use sign of larger number -18 + 8 = -10 60 ndash 20 = 40
Positive - dirt in the hole
Negative number -digging the hole
ndashPositive number dollars in your pocket Negative number dollars borrowed
Think temperature
Po
siti
ve i
s w
arm
ing
up
Neg
ative is coo
ling
off
Think moneyThink Holes
WATCH YOUR SIGNS
Multiplying or Dividing
If signs are the same answer is positive4 8 = 32 -63 -7 = -9
If signs are different answer is negative-6 7 = -42 -100 10 = -10
Distribution PrincipleMultiply everything in parenthesis by number next
to parenthesis FIRST THING
THINK BIG MOUTH SHOUTING ldquoDO MEDO ME FIRSTrdquo
EXAMPLE 3( 6x ndash 3)
3 (6x ndash 3) = 18x - 9
Distribution PrincipleMultiply everything in parenthesis by number next
to parenthesis FIRST THING
Get them off the busSo they can play football
EXAMPLE 3( 6x ndash 3)
3
18x - 9
6xndash 3
Example 2(10x ndash 3) = 6x + 2
Get the Teams off the Bus2(10x ndash 3) = 6x + 2
Line up the TeamsPenalty for off-sides ndash must change signs
Huddle up 14x = 14
Man on Man Defense 14x = 14 14 14
X = 1
Line of Scrimmage
20x ndash 6 = 6x + 8
20x ndash 6x = 8 + 6
Equation Solving
Think Football ndash Letters Vs Numbers
le rsaquo or lsaquo or geIf the equation has an Inequality sign follow the steps for solving equations with = signs
Play FootballLetters Vrs Numbers
Last Play of the Game
If you have to MULTIPLY or DIVIDE by a negative number Be sure to FLIP the Inequality
NOTICEUGLY NUMBERS WORK THE SAME AS PRETTY NUMBERS
Irsquom not afraid of
Fractions Have
Calculator Will
Calculate
UGLY Numbers Work the Same as PRETTY Numbers
If you can solve 2X + 10 = 40
Then you can solve 5x + 193 = 456
And you can solve
5x + ⅝ = ⅓
Play FootballLetters Vrs Numbers
Clue Words for writing equations from word problems
+
Word Clue
Plusadded to
the sum of increasing by
more than
Word Sentence
1 plus 56 is added to a numberThe sum of 5 and a numberA number is increased by 1015 is more than a number
Algebraic
1+56+x
5+x
x+10
x+15
Addition
Clue Words for writing equations from word problems
AlgebraicWord Clue
Minussubtracted fromthe difference ofdecreased byLessless than
Word Sentence
6 minus 57 subtracted from a numberThe difference of a number and 10A number is decreased by 205 less a number6 less than a number
6-5x-7
x-10
x-205-xx-6
Subtraction __
Clue Words for writing equations from word problems
Word Clue Word Sentence Algebraic
TimesProduct
DoubledTwiceOf (fractions and percents)
7 times a numberProduct of 8 and a numberA number doubledTwice a number12 a number55 of a number
7x = 7x8x
2x2x = 2x12x055x
Multiplication
Word Clue Word Sentence Algebraic
divide
Quotient
divided by
The quotient of a number and 710 divided by a number
xdivide7
10dividex
The first number written before the clue word will be the numerator
Clue Words for writing equations from word problems
Division
1 Consistent ndash one or many solutions2 Inconsistent ndash No solution
1 Independent ndash Only one solution2 Dependent ndash Has infinitely many
solutions
Slope Y ndash Int Graph Type of Systems
of Solutions
Same Same Same line ConsistentDependent
Infinitely many
Same Different Parallel Inconsistent 0
Different DifferentSame
Intersects ConsistentIndependent
1
Consistent and Inconsistent Systems
Slope ndash Intercept Form
y = mx + b
Slope- directions
RiseRun
Y Intercept ndash where to start
Itrsquos a line address
If the slope is a whole number put it on a stick m = 2 slope is 21
To Graph
y = 2X + 1 Starts at 1
Riserun = 21
Directions are up 2 over 1
Example 1 Example 2y = -3X+ 0
y = -3X
Starts at 0
riserun = 3-1
Directions are up 3 over -1
Thanks to httpwwwmathsisfuncomequation_of_linehtml
Linear Equations Standard Form ax + by = cSolving for y Just 3 easy stepsGreenisms Math Terms
1Move x term(Change sides Change signs) AddSubtract x term 2Give x side a hug Parenthesis ( )3Divide by number next to the y Coefficient
Example Solve for Y2x ndash 7y = 12Just 3 easy steps1 -7y = 12 ndash 2x (Move x term (Change sides Change signs)2 -7y = (12-2x) (Give it a hug)3 y = (12-2x) -7 (Divide by number next to the y)
Now you are ready to enter it into the calculator and graph it
WATCH YOUR SIGNS
Example Solve for Y2x ndash 7y = 12
Just 3 easy steps1 -7y = 12 ndash 2x X is offside Penalty change signs2 -7y = (12-2x) Huddle up ( )3 y = (12-2x) -7 Man on man defense
WATCH YOUR SIGNS
Linear Equations Standard Form ax + by = cSolving for y Itrsquos a Football Game
Y VS Everybody Else
Follow football rules
Play FootballY vs everybody else
Now you are ready to enter it into the calculator and graph it
l lines
Same Slope
Slopes are Negative Reciprocal
(Flip amp Change Sign)
PARALLEL LINES
y2 ndash y1
x2 ndash x1
or
y = mx + b
slope
Find Equation of the Line y = mx + b
I need slope (m) amp the y-intercept
(b)
MY ANSWER
y = x +
To find m ndash Solve the equation for y and use mor use the y2 ndash y1
x2 ndash x1 formula
To find b - Plug x y and m into the line equation and solve for b
(Just follow the Rules)
Base xsup2Exponent
Like Bases Exponent Example
Multiply Add
Power up Multiply
Divide Subtract s
Add No change 3asup2 + 5asup2 = 8asup2 asup2 + = asup2 +
Subtract No change 10asup2 - 4asup2 = 6asup2
5a
4
15 9 3a g y
6 4 5
3 7 4a x ka x k
5a
5a
5a
Exponents
=
(xsup2asup3g)(xasup2gsup3) = (xsup3 g )
( gsup3 y) sup3 =
asup3k xsup3
Negative ExponentsJust switch places and make exponent positive
5x5
1x
Example Switch Simplify
5 4 3
2 2 4x g ka x k
4 2
2 5 4 3g x
a x k k
4
2 3 7g
a x k
=
Xsup2
X X∙ = Xsup2 X
X times X is X squared
X
QuadraticsMultiplying Binomials ndash Draw the Face
(x + 8) (x ndash 6)
Multiply Watch your Signs
xsup2 +8x -6x -48
Simplify (Combine Like Terms eat the buggers)xsup2 + 2x -48
Check This First
The Math ldquoFrdquo Word ldquoFactoringrdquo
1 Is there a common factor (number or letter)
NO
Proceed to question 2
Yes
Proceed with CGF
4xsup3 2 2 x x x
4x ( 2xy + xsup2 -3 )
Example
8xsup2y + 4xsup3 - 12x
Factor each term8xsup2y 2 2 2 x x y 12x 2 2 3 x
Circle common terms
8xsup2y 2 2 2 x x y 4xsup3 2 2 x x x 12x 2 2 3 x
Multiply circled numbersThatrsquos your Common Factor
Multiply leftovers put in ( )
2 Perfect squares on end amp 3 terms
xsup2 - 10x + 25
YES SPLIT IT NICE
NO proceed to question 3
xsup2 - 10x + 25Put out baggies to hold the answer (parenthesis)
( x - 5 ) ( x - 5 )
Split the first term nice
Place in baggies
Sign between is same as middle term
Split the second term nice
Place in baggies
ANSWER
(x -5)sup2
3 Perfect squares on end amp 2 terms
YesSame as question 2Except the signsare +-
Example
xsup2 - 64
NO proceed to question 4
( x - 8 ) ( x + 8)
Answer looks like this
4 No perfect squares on end 3 terms amp starts with xsup2
NO proceed to question 5
YES its quadratics in the morning
xsup2 - 10x + 24MA
Multiply to +24 Add to -10
38 11
-3 -8 -11
-2 -12 -14
-6-4 -10 Found it
Put in the parenthesis
(x-6) (x-4)
All Done
2 Search and Seizure (quadratics in the morning)(See question 4)
1 Steal the ldquoardquo and give it to the last term (Multiply)1
5 Is there a number in front of the xsup2 amp does it have 3 terms
Yes Jail BreakNO
Then it is Prime
(canrsquot factor)
Example 2xsup2 + 7x + 3
xsup2 + 7x + 6 (23)
( x + 6 ) ( x + 1 )
3 Arrested and Caught - Divide last terms by ldquoardquo
( x + 62 ) ( x + 12 )
4 Beat it Down - (reduce fractions)
( x + 3 ) ( x + 12 )
5 Parole (kick denominator to the front)(x + 3 ) ( 2 x + 1 )
All Done
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Check This First
Is there a common factor (number or letter)
Greatest Common Factor (GCF)
1 Factor each term2 Circle common terms3 Multiply common terms4 Write it down5 Put out baggie for leftover6 Multiply leftovers for each
term7 Put in baggie ( )
Example
8xsup2y + 4xsup3 - 12x
4x ( 2xy + xsup2 - 3 )
Is there a Square on each End
Perfect Squares
1 Put out Parenthesis2 Split FIRST and LAST numbers nice3 Put in Signs
3 Different Kinds
A xsup2 - 16x + 64
(xndash 8) (x ndash 8)
B xsup2 + 18x + 81
(x + 9) (x + 9)
C xsup2 - 36
(x + 6) (x ndash 6)
Is there a number in front of the xsup2 and does it have 3 terms
Jail Break
1Steal the ldquoardquo and give it to the last term (multiply)2Search and Seizure (Quad in AM)3Arrested and Caught (divide by ldquoardquo)4Beat Down (reduce fractions)5Parole (kick denominator to the front)6Check it out (FACE it)
Example
2xsup2 + 7x + 3
1xsup2 + 7x + 62(x + 6) (x + 1)3(x + 62) (x + 12)4(x + 3) (x +12)5(x + 3) (2x + 1)
No perfect squares and 3 terms and starts with xsup2
Quadratics in the Morning (AM)
1 Make a factor tree2 Multiply to last number add
to middle number3 Put out baggies (parenthesis)4 Split first term nice5 Drop in factors from tree
Example
xsup2 + 12x + 32
(x + 8) (x + 4)
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Solve with Graphing Calculator
1) Solve each equation for y (3 easy steps)
2) Use y= button and enter each equation
3) Use graph to eyeball answerOr
4) Use to find where y1 and y2 are equal
5) Be sure answer is in (x y) form
Y=
Systems of EquationsTo Solve by Graphing
1) Make an x y Chart
2) Select any x solve for y x 2x ndash 4 y 0 2(0) ndash 4 -4 1 2(1) ndash 4 -2
3) Then graph the two points4) Do for both equations5) The answer is where they
cross6) Be sure answer is in (x y)
form
2nd TABLE
bull y + 2x = 9
bull y = 9 ndash 2xbull 3(9 - 2x) ndash 2x = 11
bull 27-6x ndash 2x = 11bull 27 ndash 8x = 11bull - 8x = -16bull x = 2bull y = 9 ndash 2 (2)bull y = 5bull (2 5)
Systems of EquationsSolve by Substitution
(box amp shove)
1) Solve one equation for x or y (change sides change signs)
2) Box it3) Rewrite other equation
and shove box in 4) Solve for surviving letter
1) Distribute2) Combine like terms3) Solve
4) Send it back to box5) Solve for other letter6) Answer in (xy) form
3y -2x = 11y + 2x = 9
Systems of EquationsSolve by Elimination
1) Look for opposite signs
2) Multiply to create opposites
3) Add old and new equation together
4) Solve for surviving letter
5) Plug back into either equation (pick the easy one)
6) Solve for other letter
7) Answer in (xy) format
bull 2x ndash y = 93x + 4y = -14
bull 4(2x ndash y = 9)
bull 8x ndash 4y = 36bull 3x + 4y = -14bull 11x = 22bull x = 2
bull 2(2) ndash y = 9bull -y = 5bull y = -5
bull (2 -5)
2x ndash y = 93x + 4y = -14
Do the Easy Ones First Then go Back and do the Hard Ones
For Multiple Choice TestsbullCheck each answer ndash if impossible or silly cross it out
bullBack plug (substitute) ndash one of them has to be the answer
bullFor factoring ndash Work the problem backwards
bullSketch a picture
bullGraph the points
bullUse the y= function on calculator to match graphs
Read the directions
for the test carefully
Read each question carefully
FAQs
Number next to a Letter (variable) means Multiply
4x
If x = -3
Then substitute amp multiply
4(-3) = -12
12a + -2b
If a = 5 b = -3
Then substitute amp multiply
12(5) + -2(-3) = 66
So do numbers next to ( ) and letters next to letters xy = xy
4(x) = 4x
(-2)(a) = -2aab = ab
4f g = 4fg
Addition and Subtraction are snobs They just combine with their own kind They form cliques
3x ndash 3a + 7 + 6x + 2a = 9x ndash a + 7
4xsup2 + 6x +9 ndash 15x + 3xsup2 + 10 = 7xsup2 - 9x + 19
Thatrsquos Just How They Do Thatrsquos How It Is
Deal With It
Multiplication and Division are party animals They will do it with anyone2x 4y = 8xy a bc = abc 12xy4x = 3y
Understanding Algebraic Culture is the Key to Success
23 = 2 divide 3
Numbers come with Signs (+ -)The sign is in front
Remember These are the same thing
5 ndash 3 = 25 + - 3 = 2
Because Subtraction is Adding the Opposite
If you are confused Circle the number amp the sign in front then do the math
2 ndash 56 + 7 ndash 8 ndash 10 = - 65
ABSOLUTE VALUE
The absolute value is always positive
The absolute value of 5 is 5The absolute value of -5 is 5
To solve drop the bars and make the inside number positive
Watch Out -6 = - 6 because the negative is lurking outside the bars
ALGEBRA OPPOSITES
Opposite of Multiplication is Division
25 4 = 100 100 divide 4 = 25
Positive Negative Numbers
The opposite of -5 is 5
The opposite of 6 is -6
Opposites add to zero
-4 + 4 = 0
Opposite of Addition is
Subtraction12 + 18 = 30
30 ndash 18 = 12
Opposite of Squaring is Square Rooting = 25 = 5
Adding Negative numbers - Think MONEY $$$$$$ You wonrsquot miss it
Algebra Truths
UGLY numbers work the
same as PRETTY Numbers
You canrsquot add FROGS amp
SMILEY FACES
LETTERS amp NUMBERS
work the same
Simplifying is cleaning up
your room put all the
FROGS together amp all the
SMILEY FACES together
then do the Math
No Equal Sign Simplify It Combine Like TermsSimplifying is like cleaning up your room put all the FROGS (Variables) together and all the SMILEY FACES (Numbers) together then do the Math Donrsquot Forget Numbers come with Signs
4x -5 +6x +21 -8x
4x-5
+6x+21
-8x
--------------------------------------------------------------------------------------------------------------------------------4x +6x -8x -5 +21
4x +6x -8x-5 +21
-------------------------------------------------------------------------------------------------------------------------------- 2x +16
2x +16
You canrsquot add FROGS amp SMILEY FACES so you are finished Good Job
WATCH YOUR SIGNS
Adding or Subtracting
If signs are the same add them and use the sign45 + 34 = 79 - 45 ndash 10 = -55
If signs are different subtract and use sign of larger number -18 + 8 = -10 60 ndash 20 = 40
Positive - dirt in the hole
Negative number -digging the hole
ndashPositive number dollars in your pocket Negative number dollars borrowed
Think temperature
Po
siti
ve i
s w
arm
ing
up
Neg
ative is coo
ling
off
Think moneyThink Holes
WATCH YOUR SIGNS
Multiplying or Dividing
If signs are the same answer is positive4 8 = 32 -63 -7 = -9
If signs are different answer is negative-6 7 = -42 -100 10 = -10
Distribution PrincipleMultiply everything in parenthesis by number next
to parenthesis FIRST THING
THINK BIG MOUTH SHOUTING ldquoDO MEDO ME FIRSTrdquo
EXAMPLE 3( 6x ndash 3)
3 (6x ndash 3) = 18x - 9
Distribution PrincipleMultiply everything in parenthesis by number next
to parenthesis FIRST THING
Get them off the busSo they can play football
EXAMPLE 3( 6x ndash 3)
3
18x - 9
6xndash 3
Example 2(10x ndash 3) = 6x + 2
Get the Teams off the Bus2(10x ndash 3) = 6x + 2
Line up the TeamsPenalty for off-sides ndash must change signs
Huddle up 14x = 14
Man on Man Defense 14x = 14 14 14
X = 1
Line of Scrimmage
20x ndash 6 = 6x + 8
20x ndash 6x = 8 + 6
Equation Solving
Think Football ndash Letters Vs Numbers
le rsaquo or lsaquo or geIf the equation has an Inequality sign follow the steps for solving equations with = signs
Play FootballLetters Vrs Numbers
Last Play of the Game
If you have to MULTIPLY or DIVIDE by a negative number Be sure to FLIP the Inequality
NOTICEUGLY NUMBERS WORK THE SAME AS PRETTY NUMBERS
Irsquom not afraid of
Fractions Have
Calculator Will
Calculate
UGLY Numbers Work the Same as PRETTY Numbers
If you can solve 2X + 10 = 40
Then you can solve 5x + 193 = 456
And you can solve
5x + ⅝ = ⅓
Play FootballLetters Vrs Numbers
Clue Words for writing equations from word problems
+
Word Clue
Plusadded to
the sum of increasing by
more than
Word Sentence
1 plus 56 is added to a numberThe sum of 5 and a numberA number is increased by 1015 is more than a number
Algebraic
1+56+x
5+x
x+10
x+15
Addition
Clue Words for writing equations from word problems
AlgebraicWord Clue
Minussubtracted fromthe difference ofdecreased byLessless than
Word Sentence
6 minus 57 subtracted from a numberThe difference of a number and 10A number is decreased by 205 less a number6 less than a number
6-5x-7
x-10
x-205-xx-6
Subtraction __
Clue Words for writing equations from word problems
Word Clue Word Sentence Algebraic
TimesProduct
DoubledTwiceOf (fractions and percents)
7 times a numberProduct of 8 and a numberA number doubledTwice a number12 a number55 of a number
7x = 7x8x
2x2x = 2x12x055x
Multiplication
Word Clue Word Sentence Algebraic
divide
Quotient
divided by
The quotient of a number and 710 divided by a number
xdivide7
10dividex
The first number written before the clue word will be the numerator
Clue Words for writing equations from word problems
Division
1 Consistent ndash one or many solutions2 Inconsistent ndash No solution
1 Independent ndash Only one solution2 Dependent ndash Has infinitely many
solutions
Slope Y ndash Int Graph Type of Systems
of Solutions
Same Same Same line ConsistentDependent
Infinitely many
Same Different Parallel Inconsistent 0
Different DifferentSame
Intersects ConsistentIndependent
1
Consistent and Inconsistent Systems
Slope ndash Intercept Form
y = mx + b
Slope- directions
RiseRun
Y Intercept ndash where to start
Itrsquos a line address
If the slope is a whole number put it on a stick m = 2 slope is 21
To Graph
y = 2X + 1 Starts at 1
Riserun = 21
Directions are up 2 over 1
Example 1 Example 2y = -3X+ 0
y = -3X
Starts at 0
riserun = 3-1
Directions are up 3 over -1
Thanks to httpwwwmathsisfuncomequation_of_linehtml
Linear Equations Standard Form ax + by = cSolving for y Just 3 easy stepsGreenisms Math Terms
1Move x term(Change sides Change signs) AddSubtract x term 2Give x side a hug Parenthesis ( )3Divide by number next to the y Coefficient
Example Solve for Y2x ndash 7y = 12Just 3 easy steps1 -7y = 12 ndash 2x (Move x term (Change sides Change signs)2 -7y = (12-2x) (Give it a hug)3 y = (12-2x) -7 (Divide by number next to the y)
Now you are ready to enter it into the calculator and graph it
WATCH YOUR SIGNS
Example Solve for Y2x ndash 7y = 12
Just 3 easy steps1 -7y = 12 ndash 2x X is offside Penalty change signs2 -7y = (12-2x) Huddle up ( )3 y = (12-2x) -7 Man on man defense
WATCH YOUR SIGNS
Linear Equations Standard Form ax + by = cSolving for y Itrsquos a Football Game
Y VS Everybody Else
Follow football rules
Play FootballY vs everybody else
Now you are ready to enter it into the calculator and graph it
l lines
Same Slope
Slopes are Negative Reciprocal
(Flip amp Change Sign)
PARALLEL LINES
y2 ndash y1
x2 ndash x1
or
y = mx + b
slope
Find Equation of the Line y = mx + b
I need slope (m) amp the y-intercept
(b)
MY ANSWER
y = x +
To find m ndash Solve the equation for y and use mor use the y2 ndash y1
x2 ndash x1 formula
To find b - Plug x y and m into the line equation and solve for b
(Just follow the Rules)
Base xsup2Exponent
Like Bases Exponent Example
Multiply Add
Power up Multiply
Divide Subtract s
Add No change 3asup2 + 5asup2 = 8asup2 asup2 + = asup2 +
Subtract No change 10asup2 - 4asup2 = 6asup2
5a
4
15 9 3a g y
6 4 5
3 7 4a x ka x k
5a
5a
5a
Exponents
=
(xsup2asup3g)(xasup2gsup3) = (xsup3 g )
( gsup3 y) sup3 =
asup3k xsup3
Negative ExponentsJust switch places and make exponent positive
5x5
1x
Example Switch Simplify
5 4 3
2 2 4x g ka x k
4 2
2 5 4 3g x
a x k k
4
2 3 7g
a x k
=
Xsup2
X X∙ = Xsup2 X
X times X is X squared
X
QuadraticsMultiplying Binomials ndash Draw the Face
(x + 8) (x ndash 6)
Multiply Watch your Signs
xsup2 +8x -6x -48
Simplify (Combine Like Terms eat the buggers)xsup2 + 2x -48
Check This First
The Math ldquoFrdquo Word ldquoFactoringrdquo
1 Is there a common factor (number or letter)
NO
Proceed to question 2
Yes
Proceed with CGF
4xsup3 2 2 x x x
4x ( 2xy + xsup2 -3 )
Example
8xsup2y + 4xsup3 - 12x
Factor each term8xsup2y 2 2 2 x x y 12x 2 2 3 x
Circle common terms
8xsup2y 2 2 2 x x y 4xsup3 2 2 x x x 12x 2 2 3 x
Multiply circled numbersThatrsquos your Common Factor
Multiply leftovers put in ( )
2 Perfect squares on end amp 3 terms
xsup2 - 10x + 25
YES SPLIT IT NICE
NO proceed to question 3
xsup2 - 10x + 25Put out baggies to hold the answer (parenthesis)
( x - 5 ) ( x - 5 )
Split the first term nice
Place in baggies
Sign between is same as middle term
Split the second term nice
Place in baggies
ANSWER
(x -5)sup2
3 Perfect squares on end amp 2 terms
YesSame as question 2Except the signsare +-
Example
xsup2 - 64
NO proceed to question 4
( x - 8 ) ( x + 8)
Answer looks like this
4 No perfect squares on end 3 terms amp starts with xsup2
NO proceed to question 5
YES its quadratics in the morning
xsup2 - 10x + 24MA
Multiply to +24 Add to -10
38 11
-3 -8 -11
-2 -12 -14
-6-4 -10 Found it
Put in the parenthesis
(x-6) (x-4)
All Done
2 Search and Seizure (quadratics in the morning)(See question 4)
1 Steal the ldquoardquo and give it to the last term (Multiply)1
5 Is there a number in front of the xsup2 amp does it have 3 terms
Yes Jail BreakNO
Then it is Prime
(canrsquot factor)
Example 2xsup2 + 7x + 3
xsup2 + 7x + 6 (23)
( x + 6 ) ( x + 1 )
3 Arrested and Caught - Divide last terms by ldquoardquo
( x + 62 ) ( x + 12 )
4 Beat it Down - (reduce fractions)
( x + 3 ) ( x + 12 )
5 Parole (kick denominator to the front)(x + 3 ) ( 2 x + 1 )
All Done
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Check This First
Is there a common factor (number or letter)
Greatest Common Factor (GCF)
1 Factor each term2 Circle common terms3 Multiply common terms4 Write it down5 Put out baggie for leftover6 Multiply leftovers for each
term7 Put in baggie ( )
Example
8xsup2y + 4xsup3 - 12x
4x ( 2xy + xsup2 - 3 )
Is there a Square on each End
Perfect Squares
1 Put out Parenthesis2 Split FIRST and LAST numbers nice3 Put in Signs
3 Different Kinds
A xsup2 - 16x + 64
(xndash 8) (x ndash 8)
B xsup2 + 18x + 81
(x + 9) (x + 9)
C xsup2 - 36
(x + 6) (x ndash 6)
Is there a number in front of the xsup2 and does it have 3 terms
Jail Break
1Steal the ldquoardquo and give it to the last term (multiply)2Search and Seizure (Quad in AM)3Arrested and Caught (divide by ldquoardquo)4Beat Down (reduce fractions)5Parole (kick denominator to the front)6Check it out (FACE it)
Example
2xsup2 + 7x + 3
1xsup2 + 7x + 62(x + 6) (x + 1)3(x + 62) (x + 12)4(x + 3) (x +12)5(x + 3) (2x + 1)
No perfect squares and 3 terms and starts with xsup2
Quadratics in the Morning (AM)
1 Make a factor tree2 Multiply to last number add
to middle number3 Put out baggies (parenthesis)4 Split first term nice5 Drop in factors from tree
Example
xsup2 + 12x + 32
(x + 8) (x + 4)
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Solve with Graphing Calculator
1) Solve each equation for y (3 easy steps)
2) Use y= button and enter each equation
3) Use graph to eyeball answerOr
4) Use to find where y1 and y2 are equal
5) Be sure answer is in (x y) form
Y=
Systems of EquationsTo Solve by Graphing
1) Make an x y Chart
2) Select any x solve for y x 2x ndash 4 y 0 2(0) ndash 4 -4 1 2(1) ndash 4 -2
3) Then graph the two points4) Do for both equations5) The answer is where they
cross6) Be sure answer is in (x y)
form
2nd TABLE
bull y + 2x = 9
bull y = 9 ndash 2xbull 3(9 - 2x) ndash 2x = 11
bull 27-6x ndash 2x = 11bull 27 ndash 8x = 11bull - 8x = -16bull x = 2bull y = 9 ndash 2 (2)bull y = 5bull (2 5)
Systems of EquationsSolve by Substitution
(box amp shove)
1) Solve one equation for x or y (change sides change signs)
2) Box it3) Rewrite other equation
and shove box in 4) Solve for surviving letter
1) Distribute2) Combine like terms3) Solve
4) Send it back to box5) Solve for other letter6) Answer in (xy) form
3y -2x = 11y + 2x = 9
Systems of EquationsSolve by Elimination
1) Look for opposite signs
2) Multiply to create opposites
3) Add old and new equation together
4) Solve for surviving letter
5) Plug back into either equation (pick the easy one)
6) Solve for other letter
7) Answer in (xy) format
bull 2x ndash y = 93x + 4y = -14
bull 4(2x ndash y = 9)
bull 8x ndash 4y = 36bull 3x + 4y = -14bull 11x = 22bull x = 2
bull 2(2) ndash y = 9bull -y = 5bull y = -5
bull (2 -5)
2x ndash y = 93x + 4y = -14
For Multiple Choice TestsbullCheck each answer ndash if impossible or silly cross it out
bullBack plug (substitute) ndash one of them has to be the answer
bullFor factoring ndash Work the problem backwards
bullSketch a picture
bullGraph the points
bullUse the y= function on calculator to match graphs
Read the directions
for the test carefully
Read each question carefully
FAQs
Number next to a Letter (variable) means Multiply
4x
If x = -3
Then substitute amp multiply
4(-3) = -12
12a + -2b
If a = 5 b = -3
Then substitute amp multiply
12(5) + -2(-3) = 66
So do numbers next to ( ) and letters next to letters xy = xy
4(x) = 4x
(-2)(a) = -2aab = ab
4f g = 4fg
Addition and Subtraction are snobs They just combine with their own kind They form cliques
3x ndash 3a + 7 + 6x + 2a = 9x ndash a + 7
4xsup2 + 6x +9 ndash 15x + 3xsup2 + 10 = 7xsup2 - 9x + 19
Thatrsquos Just How They Do Thatrsquos How It Is
Deal With It
Multiplication and Division are party animals They will do it with anyone2x 4y = 8xy a bc = abc 12xy4x = 3y
Understanding Algebraic Culture is the Key to Success
23 = 2 divide 3
Numbers come with Signs (+ -)The sign is in front
Remember These are the same thing
5 ndash 3 = 25 + - 3 = 2
Because Subtraction is Adding the Opposite
If you are confused Circle the number amp the sign in front then do the math
2 ndash 56 + 7 ndash 8 ndash 10 = - 65
ABSOLUTE VALUE
The absolute value is always positive
The absolute value of 5 is 5The absolute value of -5 is 5
To solve drop the bars and make the inside number positive
Watch Out -6 = - 6 because the negative is lurking outside the bars
ALGEBRA OPPOSITES
Opposite of Multiplication is Division
25 4 = 100 100 divide 4 = 25
Positive Negative Numbers
The opposite of -5 is 5
The opposite of 6 is -6
Opposites add to zero
-4 + 4 = 0
Opposite of Addition is
Subtraction12 + 18 = 30
30 ndash 18 = 12
Opposite of Squaring is Square Rooting = 25 = 5
Adding Negative numbers - Think MONEY $$$$$$ You wonrsquot miss it
Algebra Truths
UGLY numbers work the
same as PRETTY Numbers
You canrsquot add FROGS amp
SMILEY FACES
LETTERS amp NUMBERS
work the same
Simplifying is cleaning up
your room put all the
FROGS together amp all the
SMILEY FACES together
then do the Math
No Equal Sign Simplify It Combine Like TermsSimplifying is like cleaning up your room put all the FROGS (Variables) together and all the SMILEY FACES (Numbers) together then do the Math Donrsquot Forget Numbers come with Signs
4x -5 +6x +21 -8x
4x-5
+6x+21
-8x
--------------------------------------------------------------------------------------------------------------------------------4x +6x -8x -5 +21
4x +6x -8x-5 +21
-------------------------------------------------------------------------------------------------------------------------------- 2x +16
2x +16
You canrsquot add FROGS amp SMILEY FACES so you are finished Good Job
WATCH YOUR SIGNS
Adding or Subtracting
If signs are the same add them and use the sign45 + 34 = 79 - 45 ndash 10 = -55
If signs are different subtract and use sign of larger number -18 + 8 = -10 60 ndash 20 = 40
Positive - dirt in the hole
Negative number -digging the hole
ndashPositive number dollars in your pocket Negative number dollars borrowed
Think temperature
Po
siti
ve i
s w
arm
ing
up
Neg
ative is coo
ling
off
Think moneyThink Holes
WATCH YOUR SIGNS
Multiplying or Dividing
If signs are the same answer is positive4 8 = 32 -63 -7 = -9
If signs are different answer is negative-6 7 = -42 -100 10 = -10
Distribution PrincipleMultiply everything in parenthesis by number next
to parenthesis FIRST THING
THINK BIG MOUTH SHOUTING ldquoDO MEDO ME FIRSTrdquo
EXAMPLE 3( 6x ndash 3)
3 (6x ndash 3) = 18x - 9
Distribution PrincipleMultiply everything in parenthesis by number next
to parenthesis FIRST THING
Get them off the busSo they can play football
EXAMPLE 3( 6x ndash 3)
3
18x - 9
6xndash 3
Example 2(10x ndash 3) = 6x + 2
Get the Teams off the Bus2(10x ndash 3) = 6x + 2
Line up the TeamsPenalty for off-sides ndash must change signs
Huddle up 14x = 14
Man on Man Defense 14x = 14 14 14
X = 1
Line of Scrimmage
20x ndash 6 = 6x + 8
20x ndash 6x = 8 + 6
Equation Solving
Think Football ndash Letters Vs Numbers
le rsaquo or lsaquo or geIf the equation has an Inequality sign follow the steps for solving equations with = signs
Play FootballLetters Vrs Numbers
Last Play of the Game
If you have to MULTIPLY or DIVIDE by a negative number Be sure to FLIP the Inequality
NOTICEUGLY NUMBERS WORK THE SAME AS PRETTY NUMBERS
Irsquom not afraid of
Fractions Have
Calculator Will
Calculate
UGLY Numbers Work the Same as PRETTY Numbers
If you can solve 2X + 10 = 40
Then you can solve 5x + 193 = 456
And you can solve
5x + ⅝ = ⅓
Play FootballLetters Vrs Numbers
Clue Words for writing equations from word problems
+
Word Clue
Plusadded to
the sum of increasing by
more than
Word Sentence
1 plus 56 is added to a numberThe sum of 5 and a numberA number is increased by 1015 is more than a number
Algebraic
1+56+x
5+x
x+10
x+15
Addition
Clue Words for writing equations from word problems
AlgebraicWord Clue
Minussubtracted fromthe difference ofdecreased byLessless than
Word Sentence
6 minus 57 subtracted from a numberThe difference of a number and 10A number is decreased by 205 less a number6 less than a number
6-5x-7
x-10
x-205-xx-6
Subtraction __
Clue Words for writing equations from word problems
Word Clue Word Sentence Algebraic
TimesProduct
DoubledTwiceOf (fractions and percents)
7 times a numberProduct of 8 and a numberA number doubledTwice a number12 a number55 of a number
7x = 7x8x
2x2x = 2x12x055x
Multiplication
Word Clue Word Sentence Algebraic
divide
Quotient
divided by
The quotient of a number and 710 divided by a number
xdivide7
10dividex
The first number written before the clue word will be the numerator
Clue Words for writing equations from word problems
Division
1 Consistent ndash one or many solutions2 Inconsistent ndash No solution
1 Independent ndash Only one solution2 Dependent ndash Has infinitely many
solutions
Slope Y ndash Int Graph Type of Systems
of Solutions
Same Same Same line ConsistentDependent
Infinitely many
Same Different Parallel Inconsistent 0
Different DifferentSame
Intersects ConsistentIndependent
1
Consistent and Inconsistent Systems
Slope ndash Intercept Form
y = mx + b
Slope- directions
RiseRun
Y Intercept ndash where to start
Itrsquos a line address
If the slope is a whole number put it on a stick m = 2 slope is 21
To Graph
y = 2X + 1 Starts at 1
Riserun = 21
Directions are up 2 over 1
Example 1 Example 2y = -3X+ 0
y = -3X
Starts at 0
riserun = 3-1
Directions are up 3 over -1
Thanks to httpwwwmathsisfuncomequation_of_linehtml
Linear Equations Standard Form ax + by = cSolving for y Just 3 easy stepsGreenisms Math Terms
1Move x term(Change sides Change signs) AddSubtract x term 2Give x side a hug Parenthesis ( )3Divide by number next to the y Coefficient
Example Solve for Y2x ndash 7y = 12Just 3 easy steps1 -7y = 12 ndash 2x (Move x term (Change sides Change signs)2 -7y = (12-2x) (Give it a hug)3 y = (12-2x) -7 (Divide by number next to the y)
Now you are ready to enter it into the calculator and graph it
WATCH YOUR SIGNS
Example Solve for Y2x ndash 7y = 12
Just 3 easy steps1 -7y = 12 ndash 2x X is offside Penalty change signs2 -7y = (12-2x) Huddle up ( )3 y = (12-2x) -7 Man on man defense
WATCH YOUR SIGNS
Linear Equations Standard Form ax + by = cSolving for y Itrsquos a Football Game
Y VS Everybody Else
Follow football rules
Play FootballY vs everybody else
Now you are ready to enter it into the calculator and graph it
l lines
Same Slope
Slopes are Negative Reciprocal
(Flip amp Change Sign)
PARALLEL LINES
y2 ndash y1
x2 ndash x1
or
y = mx + b
slope
Find Equation of the Line y = mx + b
I need slope (m) amp the y-intercept
(b)
MY ANSWER
y = x +
To find m ndash Solve the equation for y and use mor use the y2 ndash y1
x2 ndash x1 formula
To find b - Plug x y and m into the line equation and solve for b
(Just follow the Rules)
Base xsup2Exponent
Like Bases Exponent Example
Multiply Add
Power up Multiply
Divide Subtract s
Add No change 3asup2 + 5asup2 = 8asup2 asup2 + = asup2 +
Subtract No change 10asup2 - 4asup2 = 6asup2
5a
4
15 9 3a g y
6 4 5
3 7 4a x ka x k
5a
5a
5a
Exponents
=
(xsup2asup3g)(xasup2gsup3) = (xsup3 g )
( gsup3 y) sup3 =
asup3k xsup3
Negative ExponentsJust switch places and make exponent positive
5x5
1x
Example Switch Simplify
5 4 3
2 2 4x g ka x k
4 2
2 5 4 3g x
a x k k
4
2 3 7g
a x k
=
Xsup2
X X∙ = Xsup2 X
X times X is X squared
X
QuadraticsMultiplying Binomials ndash Draw the Face
(x + 8) (x ndash 6)
Multiply Watch your Signs
xsup2 +8x -6x -48
Simplify (Combine Like Terms eat the buggers)xsup2 + 2x -48
Check This First
The Math ldquoFrdquo Word ldquoFactoringrdquo
1 Is there a common factor (number or letter)
NO
Proceed to question 2
Yes
Proceed with CGF
4xsup3 2 2 x x x
4x ( 2xy + xsup2 -3 )
Example
8xsup2y + 4xsup3 - 12x
Factor each term8xsup2y 2 2 2 x x y 12x 2 2 3 x
Circle common terms
8xsup2y 2 2 2 x x y 4xsup3 2 2 x x x 12x 2 2 3 x
Multiply circled numbersThatrsquos your Common Factor
Multiply leftovers put in ( )
2 Perfect squares on end amp 3 terms
xsup2 - 10x + 25
YES SPLIT IT NICE
NO proceed to question 3
xsup2 - 10x + 25Put out baggies to hold the answer (parenthesis)
( x - 5 ) ( x - 5 )
Split the first term nice
Place in baggies
Sign between is same as middle term
Split the second term nice
Place in baggies
ANSWER
(x -5)sup2
3 Perfect squares on end amp 2 terms
YesSame as question 2Except the signsare +-
Example
xsup2 - 64
NO proceed to question 4
( x - 8 ) ( x + 8)
Answer looks like this
4 No perfect squares on end 3 terms amp starts with xsup2
NO proceed to question 5
YES its quadratics in the morning
xsup2 - 10x + 24MA
Multiply to +24 Add to -10
38 11
-3 -8 -11
-2 -12 -14
-6-4 -10 Found it
Put in the parenthesis
(x-6) (x-4)
All Done
2 Search and Seizure (quadratics in the morning)(See question 4)
1 Steal the ldquoardquo and give it to the last term (Multiply)1
5 Is there a number in front of the xsup2 amp does it have 3 terms
Yes Jail BreakNO
Then it is Prime
(canrsquot factor)
Example 2xsup2 + 7x + 3
xsup2 + 7x + 6 (23)
( x + 6 ) ( x + 1 )
3 Arrested and Caught - Divide last terms by ldquoardquo
( x + 62 ) ( x + 12 )
4 Beat it Down - (reduce fractions)
( x + 3 ) ( x + 12 )
5 Parole (kick denominator to the front)(x + 3 ) ( 2 x + 1 )
All Done
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Check This First
Is there a common factor (number or letter)
Greatest Common Factor (GCF)
1 Factor each term2 Circle common terms3 Multiply common terms4 Write it down5 Put out baggie for leftover6 Multiply leftovers for each
term7 Put in baggie ( )
Example
8xsup2y + 4xsup3 - 12x
4x ( 2xy + xsup2 - 3 )
Is there a Square on each End
Perfect Squares
1 Put out Parenthesis2 Split FIRST and LAST numbers nice3 Put in Signs
3 Different Kinds
A xsup2 - 16x + 64
(xndash 8) (x ndash 8)
B xsup2 + 18x + 81
(x + 9) (x + 9)
C xsup2 - 36
(x + 6) (x ndash 6)
Is there a number in front of the xsup2 and does it have 3 terms
Jail Break
1Steal the ldquoardquo and give it to the last term (multiply)2Search and Seizure (Quad in AM)3Arrested and Caught (divide by ldquoardquo)4Beat Down (reduce fractions)5Parole (kick denominator to the front)6Check it out (FACE it)
Example
2xsup2 + 7x + 3
1xsup2 + 7x + 62(x + 6) (x + 1)3(x + 62) (x + 12)4(x + 3) (x +12)5(x + 3) (2x + 1)
No perfect squares and 3 terms and starts with xsup2
Quadratics in the Morning (AM)
1 Make a factor tree2 Multiply to last number add
to middle number3 Put out baggies (parenthesis)4 Split first term nice5 Drop in factors from tree
Example
xsup2 + 12x + 32
(x + 8) (x + 4)
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Solve with Graphing Calculator
1) Solve each equation for y (3 easy steps)
2) Use y= button and enter each equation
3) Use graph to eyeball answerOr
4) Use to find where y1 and y2 are equal
5) Be sure answer is in (x y) form
Y=
Systems of EquationsTo Solve by Graphing
1) Make an x y Chart
2) Select any x solve for y x 2x ndash 4 y 0 2(0) ndash 4 -4 1 2(1) ndash 4 -2
3) Then graph the two points4) Do for both equations5) The answer is where they
cross6) Be sure answer is in (x y)
form
2nd TABLE
bull y + 2x = 9
bull y = 9 ndash 2xbull 3(9 - 2x) ndash 2x = 11
bull 27-6x ndash 2x = 11bull 27 ndash 8x = 11bull - 8x = -16bull x = 2bull y = 9 ndash 2 (2)bull y = 5bull (2 5)
Systems of EquationsSolve by Substitution
(box amp shove)
1) Solve one equation for x or y (change sides change signs)
2) Box it3) Rewrite other equation
and shove box in 4) Solve for surviving letter
1) Distribute2) Combine like terms3) Solve
4) Send it back to box5) Solve for other letter6) Answer in (xy) form
3y -2x = 11y + 2x = 9
Systems of EquationsSolve by Elimination
1) Look for opposite signs
2) Multiply to create opposites
3) Add old and new equation together
4) Solve for surviving letter
5) Plug back into either equation (pick the easy one)
6) Solve for other letter
7) Answer in (xy) format
bull 2x ndash y = 93x + 4y = -14
bull 4(2x ndash y = 9)
bull 8x ndash 4y = 36bull 3x + 4y = -14bull 11x = 22bull x = 2
bull 2(2) ndash y = 9bull -y = 5bull y = -5
bull (2 -5)
2x ndash y = 93x + 4y = -14
Read the directions
for the test carefully
Read each question carefully
FAQs
Number next to a Letter (variable) means Multiply
4x
If x = -3
Then substitute amp multiply
4(-3) = -12
12a + -2b
If a = 5 b = -3
Then substitute amp multiply
12(5) + -2(-3) = 66
So do numbers next to ( ) and letters next to letters xy = xy
4(x) = 4x
(-2)(a) = -2aab = ab
4f g = 4fg
Addition and Subtraction are snobs They just combine with their own kind They form cliques
3x ndash 3a + 7 + 6x + 2a = 9x ndash a + 7
4xsup2 + 6x +9 ndash 15x + 3xsup2 + 10 = 7xsup2 - 9x + 19
Thatrsquos Just How They Do Thatrsquos How It Is
Deal With It
Multiplication and Division are party animals They will do it with anyone2x 4y = 8xy a bc = abc 12xy4x = 3y
Understanding Algebraic Culture is the Key to Success
23 = 2 divide 3
Numbers come with Signs (+ -)The sign is in front
Remember These are the same thing
5 ndash 3 = 25 + - 3 = 2
Because Subtraction is Adding the Opposite
If you are confused Circle the number amp the sign in front then do the math
2 ndash 56 + 7 ndash 8 ndash 10 = - 65
ABSOLUTE VALUE
The absolute value is always positive
The absolute value of 5 is 5The absolute value of -5 is 5
To solve drop the bars and make the inside number positive
Watch Out -6 = - 6 because the negative is lurking outside the bars
ALGEBRA OPPOSITES
Opposite of Multiplication is Division
25 4 = 100 100 divide 4 = 25
Positive Negative Numbers
The opposite of -5 is 5
The opposite of 6 is -6
Opposites add to zero
-4 + 4 = 0
Opposite of Addition is
Subtraction12 + 18 = 30
30 ndash 18 = 12
Opposite of Squaring is Square Rooting = 25 = 5
Adding Negative numbers - Think MONEY $$$$$$ You wonrsquot miss it
Algebra Truths
UGLY numbers work the
same as PRETTY Numbers
You canrsquot add FROGS amp
SMILEY FACES
LETTERS amp NUMBERS
work the same
Simplifying is cleaning up
your room put all the
FROGS together amp all the
SMILEY FACES together
then do the Math
No Equal Sign Simplify It Combine Like TermsSimplifying is like cleaning up your room put all the FROGS (Variables) together and all the SMILEY FACES (Numbers) together then do the Math Donrsquot Forget Numbers come with Signs
4x -5 +6x +21 -8x
4x-5
+6x+21
-8x
--------------------------------------------------------------------------------------------------------------------------------4x +6x -8x -5 +21
4x +6x -8x-5 +21
-------------------------------------------------------------------------------------------------------------------------------- 2x +16
2x +16
You canrsquot add FROGS amp SMILEY FACES so you are finished Good Job
WATCH YOUR SIGNS
Adding or Subtracting
If signs are the same add them and use the sign45 + 34 = 79 - 45 ndash 10 = -55
If signs are different subtract and use sign of larger number -18 + 8 = -10 60 ndash 20 = 40
Positive - dirt in the hole
Negative number -digging the hole
ndashPositive number dollars in your pocket Negative number dollars borrowed
Think temperature
Po
siti
ve i
s w
arm
ing
up
Neg
ative is coo
ling
off
Think moneyThink Holes
WATCH YOUR SIGNS
Multiplying or Dividing
If signs are the same answer is positive4 8 = 32 -63 -7 = -9
If signs are different answer is negative-6 7 = -42 -100 10 = -10
Distribution PrincipleMultiply everything in parenthesis by number next
to parenthesis FIRST THING
THINK BIG MOUTH SHOUTING ldquoDO MEDO ME FIRSTrdquo
EXAMPLE 3( 6x ndash 3)
3 (6x ndash 3) = 18x - 9
Distribution PrincipleMultiply everything in parenthesis by number next
to parenthesis FIRST THING
Get them off the busSo they can play football
EXAMPLE 3( 6x ndash 3)
3
18x - 9
6xndash 3
Example 2(10x ndash 3) = 6x + 2
Get the Teams off the Bus2(10x ndash 3) = 6x + 2
Line up the TeamsPenalty for off-sides ndash must change signs
Huddle up 14x = 14
Man on Man Defense 14x = 14 14 14
X = 1
Line of Scrimmage
20x ndash 6 = 6x + 8
20x ndash 6x = 8 + 6
Equation Solving
Think Football ndash Letters Vs Numbers
le rsaquo or lsaquo or geIf the equation has an Inequality sign follow the steps for solving equations with = signs
Play FootballLetters Vrs Numbers
Last Play of the Game
If you have to MULTIPLY or DIVIDE by a negative number Be sure to FLIP the Inequality
NOTICEUGLY NUMBERS WORK THE SAME AS PRETTY NUMBERS
Irsquom not afraid of
Fractions Have
Calculator Will
Calculate
UGLY Numbers Work the Same as PRETTY Numbers
If you can solve 2X + 10 = 40
Then you can solve 5x + 193 = 456
And you can solve
5x + ⅝ = ⅓
Play FootballLetters Vrs Numbers
Clue Words for writing equations from word problems
+
Word Clue
Plusadded to
the sum of increasing by
more than
Word Sentence
1 plus 56 is added to a numberThe sum of 5 and a numberA number is increased by 1015 is more than a number
Algebraic
1+56+x
5+x
x+10
x+15
Addition
Clue Words for writing equations from word problems
AlgebraicWord Clue
Minussubtracted fromthe difference ofdecreased byLessless than
Word Sentence
6 minus 57 subtracted from a numberThe difference of a number and 10A number is decreased by 205 less a number6 less than a number
6-5x-7
x-10
x-205-xx-6
Subtraction __
Clue Words for writing equations from word problems
Word Clue Word Sentence Algebraic
TimesProduct
DoubledTwiceOf (fractions and percents)
7 times a numberProduct of 8 and a numberA number doubledTwice a number12 a number55 of a number
7x = 7x8x
2x2x = 2x12x055x
Multiplication
Word Clue Word Sentence Algebraic
divide
Quotient
divided by
The quotient of a number and 710 divided by a number
xdivide7
10dividex
The first number written before the clue word will be the numerator
Clue Words for writing equations from word problems
Division
1 Consistent ndash one or many solutions2 Inconsistent ndash No solution
1 Independent ndash Only one solution2 Dependent ndash Has infinitely many
solutions
Slope Y ndash Int Graph Type of Systems
of Solutions
Same Same Same line ConsistentDependent
Infinitely many
Same Different Parallel Inconsistent 0
Different DifferentSame
Intersects ConsistentIndependent
1
Consistent and Inconsistent Systems
Slope ndash Intercept Form
y = mx + b
Slope- directions
RiseRun
Y Intercept ndash where to start
Itrsquos a line address
If the slope is a whole number put it on a stick m = 2 slope is 21
To Graph
y = 2X + 1 Starts at 1
Riserun = 21
Directions are up 2 over 1
Example 1 Example 2y = -3X+ 0
y = -3X
Starts at 0
riserun = 3-1
Directions are up 3 over -1
Thanks to httpwwwmathsisfuncomequation_of_linehtml
Linear Equations Standard Form ax + by = cSolving for y Just 3 easy stepsGreenisms Math Terms
1Move x term(Change sides Change signs) AddSubtract x term 2Give x side a hug Parenthesis ( )3Divide by number next to the y Coefficient
Example Solve for Y2x ndash 7y = 12Just 3 easy steps1 -7y = 12 ndash 2x (Move x term (Change sides Change signs)2 -7y = (12-2x) (Give it a hug)3 y = (12-2x) -7 (Divide by number next to the y)
Now you are ready to enter it into the calculator and graph it
WATCH YOUR SIGNS
Example Solve for Y2x ndash 7y = 12
Just 3 easy steps1 -7y = 12 ndash 2x X is offside Penalty change signs2 -7y = (12-2x) Huddle up ( )3 y = (12-2x) -7 Man on man defense
WATCH YOUR SIGNS
Linear Equations Standard Form ax + by = cSolving for y Itrsquos a Football Game
Y VS Everybody Else
Follow football rules
Play FootballY vs everybody else
Now you are ready to enter it into the calculator and graph it
l lines
Same Slope
Slopes are Negative Reciprocal
(Flip amp Change Sign)
PARALLEL LINES
y2 ndash y1
x2 ndash x1
or
y = mx + b
slope
Find Equation of the Line y = mx + b
I need slope (m) amp the y-intercept
(b)
MY ANSWER
y = x +
To find m ndash Solve the equation for y and use mor use the y2 ndash y1
x2 ndash x1 formula
To find b - Plug x y and m into the line equation and solve for b
(Just follow the Rules)
Base xsup2Exponent
Like Bases Exponent Example
Multiply Add
Power up Multiply
Divide Subtract s
Add No change 3asup2 + 5asup2 = 8asup2 asup2 + = asup2 +
Subtract No change 10asup2 - 4asup2 = 6asup2
5a
4
15 9 3a g y
6 4 5
3 7 4a x ka x k
5a
5a
5a
Exponents
=
(xsup2asup3g)(xasup2gsup3) = (xsup3 g )
( gsup3 y) sup3 =
asup3k xsup3
Negative ExponentsJust switch places and make exponent positive
5x5
1x
Example Switch Simplify
5 4 3
2 2 4x g ka x k
4 2
2 5 4 3g x
a x k k
4
2 3 7g
a x k
=
Xsup2
X X∙ = Xsup2 X
X times X is X squared
X
QuadraticsMultiplying Binomials ndash Draw the Face
(x + 8) (x ndash 6)
Multiply Watch your Signs
xsup2 +8x -6x -48
Simplify (Combine Like Terms eat the buggers)xsup2 + 2x -48
Check This First
The Math ldquoFrdquo Word ldquoFactoringrdquo
1 Is there a common factor (number or letter)
NO
Proceed to question 2
Yes
Proceed with CGF
4xsup3 2 2 x x x
4x ( 2xy + xsup2 -3 )
Example
8xsup2y + 4xsup3 - 12x
Factor each term8xsup2y 2 2 2 x x y 12x 2 2 3 x
Circle common terms
8xsup2y 2 2 2 x x y 4xsup3 2 2 x x x 12x 2 2 3 x
Multiply circled numbersThatrsquos your Common Factor
Multiply leftovers put in ( )
2 Perfect squares on end amp 3 terms
xsup2 - 10x + 25
YES SPLIT IT NICE
NO proceed to question 3
xsup2 - 10x + 25Put out baggies to hold the answer (parenthesis)
( x - 5 ) ( x - 5 )
Split the first term nice
Place in baggies
Sign between is same as middle term
Split the second term nice
Place in baggies
ANSWER
(x -5)sup2
3 Perfect squares on end amp 2 terms
YesSame as question 2Except the signsare +-
Example
xsup2 - 64
NO proceed to question 4
( x - 8 ) ( x + 8)
Answer looks like this
4 No perfect squares on end 3 terms amp starts with xsup2
NO proceed to question 5
YES its quadratics in the morning
xsup2 - 10x + 24MA
Multiply to +24 Add to -10
38 11
-3 -8 -11
-2 -12 -14
-6-4 -10 Found it
Put in the parenthesis
(x-6) (x-4)
All Done
2 Search and Seizure (quadratics in the morning)(See question 4)
1 Steal the ldquoardquo and give it to the last term (Multiply)1
5 Is there a number in front of the xsup2 amp does it have 3 terms
Yes Jail BreakNO
Then it is Prime
(canrsquot factor)
Example 2xsup2 + 7x + 3
xsup2 + 7x + 6 (23)
( x + 6 ) ( x + 1 )
3 Arrested and Caught - Divide last terms by ldquoardquo
( x + 62 ) ( x + 12 )
4 Beat it Down - (reduce fractions)
( x + 3 ) ( x + 12 )
5 Parole (kick denominator to the front)(x + 3 ) ( 2 x + 1 )
All Done
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Check This First
Is there a common factor (number or letter)
Greatest Common Factor (GCF)
1 Factor each term2 Circle common terms3 Multiply common terms4 Write it down5 Put out baggie for leftover6 Multiply leftovers for each
term7 Put in baggie ( )
Example
8xsup2y + 4xsup3 - 12x
4x ( 2xy + xsup2 - 3 )
Is there a Square on each End
Perfect Squares
1 Put out Parenthesis2 Split FIRST and LAST numbers nice3 Put in Signs
3 Different Kinds
A xsup2 - 16x + 64
(xndash 8) (x ndash 8)
B xsup2 + 18x + 81
(x + 9) (x + 9)
C xsup2 - 36
(x + 6) (x ndash 6)
Is there a number in front of the xsup2 and does it have 3 terms
Jail Break
1Steal the ldquoardquo and give it to the last term (multiply)2Search and Seizure (Quad in AM)3Arrested and Caught (divide by ldquoardquo)4Beat Down (reduce fractions)5Parole (kick denominator to the front)6Check it out (FACE it)
Example
2xsup2 + 7x + 3
1xsup2 + 7x + 62(x + 6) (x + 1)3(x + 62) (x + 12)4(x + 3) (x +12)5(x + 3) (2x + 1)
No perfect squares and 3 terms and starts with xsup2
Quadratics in the Morning (AM)
1 Make a factor tree2 Multiply to last number add
to middle number3 Put out baggies (parenthesis)4 Split first term nice5 Drop in factors from tree
Example
xsup2 + 12x + 32
(x + 8) (x + 4)
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Solve with Graphing Calculator
1) Solve each equation for y (3 easy steps)
2) Use y= button and enter each equation
3) Use graph to eyeball answerOr
4) Use to find where y1 and y2 are equal
5) Be sure answer is in (x y) form
Y=
Systems of EquationsTo Solve by Graphing
1) Make an x y Chart
2) Select any x solve for y x 2x ndash 4 y 0 2(0) ndash 4 -4 1 2(1) ndash 4 -2
3) Then graph the two points4) Do for both equations5) The answer is where they
cross6) Be sure answer is in (x y)
form
2nd TABLE
bull y + 2x = 9
bull y = 9 ndash 2xbull 3(9 - 2x) ndash 2x = 11
bull 27-6x ndash 2x = 11bull 27 ndash 8x = 11bull - 8x = -16bull x = 2bull y = 9 ndash 2 (2)bull y = 5bull (2 5)
Systems of EquationsSolve by Substitution
(box amp shove)
1) Solve one equation for x or y (change sides change signs)
2) Box it3) Rewrite other equation
and shove box in 4) Solve for surviving letter
1) Distribute2) Combine like terms3) Solve
4) Send it back to box5) Solve for other letter6) Answer in (xy) form
3y -2x = 11y + 2x = 9
Systems of EquationsSolve by Elimination
1) Look for opposite signs
2) Multiply to create opposites
3) Add old and new equation together
4) Solve for surviving letter
5) Plug back into either equation (pick the easy one)
6) Solve for other letter
7) Answer in (xy) format
bull 2x ndash y = 93x + 4y = -14
bull 4(2x ndash y = 9)
bull 8x ndash 4y = 36bull 3x + 4y = -14bull 11x = 22bull x = 2
bull 2(2) ndash y = 9bull -y = 5bull y = -5
bull (2 -5)
2x ndash y = 93x + 4y = -14
Read each question carefully
FAQs
Number next to a Letter (variable) means Multiply
4x
If x = -3
Then substitute amp multiply
4(-3) = -12
12a + -2b
If a = 5 b = -3
Then substitute amp multiply
12(5) + -2(-3) = 66
So do numbers next to ( ) and letters next to letters xy = xy
4(x) = 4x
(-2)(a) = -2aab = ab
4f g = 4fg
Addition and Subtraction are snobs They just combine with their own kind They form cliques
3x ndash 3a + 7 + 6x + 2a = 9x ndash a + 7
4xsup2 + 6x +9 ndash 15x + 3xsup2 + 10 = 7xsup2 - 9x + 19
Thatrsquos Just How They Do Thatrsquos How It Is
Deal With It
Multiplication and Division are party animals They will do it with anyone2x 4y = 8xy a bc = abc 12xy4x = 3y
Understanding Algebraic Culture is the Key to Success
23 = 2 divide 3
Numbers come with Signs (+ -)The sign is in front
Remember These are the same thing
5 ndash 3 = 25 + - 3 = 2
Because Subtraction is Adding the Opposite
If you are confused Circle the number amp the sign in front then do the math
2 ndash 56 + 7 ndash 8 ndash 10 = - 65
ABSOLUTE VALUE
The absolute value is always positive
The absolute value of 5 is 5The absolute value of -5 is 5
To solve drop the bars and make the inside number positive
Watch Out -6 = - 6 because the negative is lurking outside the bars
ALGEBRA OPPOSITES
Opposite of Multiplication is Division
25 4 = 100 100 divide 4 = 25
Positive Negative Numbers
The opposite of -5 is 5
The opposite of 6 is -6
Opposites add to zero
-4 + 4 = 0
Opposite of Addition is
Subtraction12 + 18 = 30
30 ndash 18 = 12
Opposite of Squaring is Square Rooting = 25 = 5
Adding Negative numbers - Think MONEY $$$$$$ You wonrsquot miss it
Algebra Truths
UGLY numbers work the
same as PRETTY Numbers
You canrsquot add FROGS amp
SMILEY FACES
LETTERS amp NUMBERS
work the same
Simplifying is cleaning up
your room put all the
FROGS together amp all the
SMILEY FACES together
then do the Math
No Equal Sign Simplify It Combine Like TermsSimplifying is like cleaning up your room put all the FROGS (Variables) together and all the SMILEY FACES (Numbers) together then do the Math Donrsquot Forget Numbers come with Signs
4x -5 +6x +21 -8x
4x-5
+6x+21
-8x
--------------------------------------------------------------------------------------------------------------------------------4x +6x -8x -5 +21
4x +6x -8x-5 +21
-------------------------------------------------------------------------------------------------------------------------------- 2x +16
2x +16
You canrsquot add FROGS amp SMILEY FACES so you are finished Good Job
WATCH YOUR SIGNS
Adding or Subtracting
If signs are the same add them and use the sign45 + 34 = 79 - 45 ndash 10 = -55
If signs are different subtract and use sign of larger number -18 + 8 = -10 60 ndash 20 = 40
Positive - dirt in the hole
Negative number -digging the hole
ndashPositive number dollars in your pocket Negative number dollars borrowed
Think temperature
Po
siti
ve i
s w
arm
ing
up
Neg
ative is coo
ling
off
Think moneyThink Holes
WATCH YOUR SIGNS
Multiplying or Dividing
If signs are the same answer is positive4 8 = 32 -63 -7 = -9
If signs are different answer is negative-6 7 = -42 -100 10 = -10
Distribution PrincipleMultiply everything in parenthesis by number next
to parenthesis FIRST THING
THINK BIG MOUTH SHOUTING ldquoDO MEDO ME FIRSTrdquo
EXAMPLE 3( 6x ndash 3)
3 (6x ndash 3) = 18x - 9
Distribution PrincipleMultiply everything in parenthesis by number next
to parenthesis FIRST THING
Get them off the busSo they can play football
EXAMPLE 3( 6x ndash 3)
3
18x - 9
6xndash 3
Example 2(10x ndash 3) = 6x + 2
Get the Teams off the Bus2(10x ndash 3) = 6x + 2
Line up the TeamsPenalty for off-sides ndash must change signs
Huddle up 14x = 14
Man on Man Defense 14x = 14 14 14
X = 1
Line of Scrimmage
20x ndash 6 = 6x + 8
20x ndash 6x = 8 + 6
Equation Solving
Think Football ndash Letters Vs Numbers
le rsaquo or lsaquo or geIf the equation has an Inequality sign follow the steps for solving equations with = signs
Play FootballLetters Vrs Numbers
Last Play of the Game
If you have to MULTIPLY or DIVIDE by a negative number Be sure to FLIP the Inequality
NOTICEUGLY NUMBERS WORK THE SAME AS PRETTY NUMBERS
Irsquom not afraid of
Fractions Have
Calculator Will
Calculate
UGLY Numbers Work the Same as PRETTY Numbers
If you can solve 2X + 10 = 40
Then you can solve 5x + 193 = 456
And you can solve
5x + ⅝ = ⅓
Play FootballLetters Vrs Numbers
Clue Words for writing equations from word problems
+
Word Clue
Plusadded to
the sum of increasing by
more than
Word Sentence
1 plus 56 is added to a numberThe sum of 5 and a numberA number is increased by 1015 is more than a number
Algebraic
1+56+x
5+x
x+10
x+15
Addition
Clue Words for writing equations from word problems
AlgebraicWord Clue
Minussubtracted fromthe difference ofdecreased byLessless than
Word Sentence
6 minus 57 subtracted from a numberThe difference of a number and 10A number is decreased by 205 less a number6 less than a number
6-5x-7
x-10
x-205-xx-6
Subtraction __
Clue Words for writing equations from word problems
Word Clue Word Sentence Algebraic
TimesProduct
DoubledTwiceOf (fractions and percents)
7 times a numberProduct of 8 and a numberA number doubledTwice a number12 a number55 of a number
7x = 7x8x
2x2x = 2x12x055x
Multiplication
Word Clue Word Sentence Algebraic
divide
Quotient
divided by
The quotient of a number and 710 divided by a number
xdivide7
10dividex
The first number written before the clue word will be the numerator
Clue Words for writing equations from word problems
Division
1 Consistent ndash one or many solutions2 Inconsistent ndash No solution
1 Independent ndash Only one solution2 Dependent ndash Has infinitely many
solutions
Slope Y ndash Int Graph Type of Systems
of Solutions
Same Same Same line ConsistentDependent
Infinitely many
Same Different Parallel Inconsistent 0
Different DifferentSame
Intersects ConsistentIndependent
1
Consistent and Inconsistent Systems
Slope ndash Intercept Form
y = mx + b
Slope- directions
RiseRun
Y Intercept ndash where to start
Itrsquos a line address
If the slope is a whole number put it on a stick m = 2 slope is 21
To Graph
y = 2X + 1 Starts at 1
Riserun = 21
Directions are up 2 over 1
Example 1 Example 2y = -3X+ 0
y = -3X
Starts at 0
riserun = 3-1
Directions are up 3 over -1
Thanks to httpwwwmathsisfuncomequation_of_linehtml
Linear Equations Standard Form ax + by = cSolving for y Just 3 easy stepsGreenisms Math Terms
1Move x term(Change sides Change signs) AddSubtract x term 2Give x side a hug Parenthesis ( )3Divide by number next to the y Coefficient
Example Solve for Y2x ndash 7y = 12Just 3 easy steps1 -7y = 12 ndash 2x (Move x term (Change sides Change signs)2 -7y = (12-2x) (Give it a hug)3 y = (12-2x) -7 (Divide by number next to the y)
Now you are ready to enter it into the calculator and graph it
WATCH YOUR SIGNS
Example Solve for Y2x ndash 7y = 12
Just 3 easy steps1 -7y = 12 ndash 2x X is offside Penalty change signs2 -7y = (12-2x) Huddle up ( )3 y = (12-2x) -7 Man on man defense
WATCH YOUR SIGNS
Linear Equations Standard Form ax + by = cSolving for y Itrsquos a Football Game
Y VS Everybody Else
Follow football rules
Play FootballY vs everybody else
Now you are ready to enter it into the calculator and graph it
l lines
Same Slope
Slopes are Negative Reciprocal
(Flip amp Change Sign)
PARALLEL LINES
y2 ndash y1
x2 ndash x1
or
y = mx + b
slope
Find Equation of the Line y = mx + b
I need slope (m) amp the y-intercept
(b)
MY ANSWER
y = x +
To find m ndash Solve the equation for y and use mor use the y2 ndash y1
x2 ndash x1 formula
To find b - Plug x y and m into the line equation and solve for b
(Just follow the Rules)
Base xsup2Exponent
Like Bases Exponent Example
Multiply Add
Power up Multiply
Divide Subtract s
Add No change 3asup2 + 5asup2 = 8asup2 asup2 + = asup2 +
Subtract No change 10asup2 - 4asup2 = 6asup2
5a
4
15 9 3a g y
6 4 5
3 7 4a x ka x k
5a
5a
5a
Exponents
=
(xsup2asup3g)(xasup2gsup3) = (xsup3 g )
( gsup3 y) sup3 =
asup3k xsup3
Negative ExponentsJust switch places and make exponent positive
5x5
1x
Example Switch Simplify
5 4 3
2 2 4x g ka x k
4 2
2 5 4 3g x
a x k k
4
2 3 7g
a x k
=
Xsup2
X X∙ = Xsup2 X
X times X is X squared
X
QuadraticsMultiplying Binomials ndash Draw the Face
(x + 8) (x ndash 6)
Multiply Watch your Signs
xsup2 +8x -6x -48
Simplify (Combine Like Terms eat the buggers)xsup2 + 2x -48
Check This First
The Math ldquoFrdquo Word ldquoFactoringrdquo
1 Is there a common factor (number or letter)
NO
Proceed to question 2
Yes
Proceed with CGF
4xsup3 2 2 x x x
4x ( 2xy + xsup2 -3 )
Example
8xsup2y + 4xsup3 - 12x
Factor each term8xsup2y 2 2 2 x x y 12x 2 2 3 x
Circle common terms
8xsup2y 2 2 2 x x y 4xsup3 2 2 x x x 12x 2 2 3 x
Multiply circled numbersThatrsquos your Common Factor
Multiply leftovers put in ( )
2 Perfect squares on end amp 3 terms
xsup2 - 10x + 25
YES SPLIT IT NICE
NO proceed to question 3
xsup2 - 10x + 25Put out baggies to hold the answer (parenthesis)
( x - 5 ) ( x - 5 )
Split the first term nice
Place in baggies
Sign between is same as middle term
Split the second term nice
Place in baggies
ANSWER
(x -5)sup2
3 Perfect squares on end amp 2 terms
YesSame as question 2Except the signsare +-
Example
xsup2 - 64
NO proceed to question 4
( x - 8 ) ( x + 8)
Answer looks like this
4 No perfect squares on end 3 terms amp starts with xsup2
NO proceed to question 5
YES its quadratics in the morning
xsup2 - 10x + 24MA
Multiply to +24 Add to -10
38 11
-3 -8 -11
-2 -12 -14
-6-4 -10 Found it
Put in the parenthesis
(x-6) (x-4)
All Done
2 Search and Seizure (quadratics in the morning)(See question 4)
1 Steal the ldquoardquo and give it to the last term (Multiply)1
5 Is there a number in front of the xsup2 amp does it have 3 terms
Yes Jail BreakNO
Then it is Prime
(canrsquot factor)
Example 2xsup2 + 7x + 3
xsup2 + 7x + 6 (23)
( x + 6 ) ( x + 1 )
3 Arrested and Caught - Divide last terms by ldquoardquo
( x + 62 ) ( x + 12 )
4 Beat it Down - (reduce fractions)
( x + 3 ) ( x + 12 )
5 Parole (kick denominator to the front)(x + 3 ) ( 2 x + 1 )
All Done
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Check This First
Is there a common factor (number or letter)
Greatest Common Factor (GCF)
1 Factor each term2 Circle common terms3 Multiply common terms4 Write it down5 Put out baggie for leftover6 Multiply leftovers for each
term7 Put in baggie ( )
Example
8xsup2y + 4xsup3 - 12x
4x ( 2xy + xsup2 - 3 )
Is there a Square on each End
Perfect Squares
1 Put out Parenthesis2 Split FIRST and LAST numbers nice3 Put in Signs
3 Different Kinds
A xsup2 - 16x + 64
(xndash 8) (x ndash 8)
B xsup2 + 18x + 81
(x + 9) (x + 9)
C xsup2 - 36
(x + 6) (x ndash 6)
Is there a number in front of the xsup2 and does it have 3 terms
Jail Break
1Steal the ldquoardquo and give it to the last term (multiply)2Search and Seizure (Quad in AM)3Arrested and Caught (divide by ldquoardquo)4Beat Down (reduce fractions)5Parole (kick denominator to the front)6Check it out (FACE it)
Example
2xsup2 + 7x + 3
1xsup2 + 7x + 62(x + 6) (x + 1)3(x + 62) (x + 12)4(x + 3) (x +12)5(x + 3) (2x + 1)
No perfect squares and 3 terms and starts with xsup2
Quadratics in the Morning (AM)
1 Make a factor tree2 Multiply to last number add
to middle number3 Put out baggies (parenthesis)4 Split first term nice5 Drop in factors from tree
Example
xsup2 + 12x + 32
(x + 8) (x + 4)
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Solve with Graphing Calculator
1) Solve each equation for y (3 easy steps)
2) Use y= button and enter each equation
3) Use graph to eyeball answerOr
4) Use to find where y1 and y2 are equal
5) Be sure answer is in (x y) form
Y=
Systems of EquationsTo Solve by Graphing
1) Make an x y Chart
2) Select any x solve for y x 2x ndash 4 y 0 2(0) ndash 4 -4 1 2(1) ndash 4 -2
3) Then graph the two points4) Do for both equations5) The answer is where they
cross6) Be sure answer is in (x y)
form
2nd TABLE
bull y + 2x = 9
bull y = 9 ndash 2xbull 3(9 - 2x) ndash 2x = 11
bull 27-6x ndash 2x = 11bull 27 ndash 8x = 11bull - 8x = -16bull x = 2bull y = 9 ndash 2 (2)bull y = 5bull (2 5)
Systems of EquationsSolve by Substitution
(box amp shove)
1) Solve one equation for x or y (change sides change signs)
2) Box it3) Rewrite other equation
and shove box in 4) Solve for surviving letter
1) Distribute2) Combine like terms3) Solve
4) Send it back to box5) Solve for other letter6) Answer in (xy) form
3y -2x = 11y + 2x = 9
Systems of EquationsSolve by Elimination
1) Look for opposite signs
2) Multiply to create opposites
3) Add old and new equation together
4) Solve for surviving letter
5) Plug back into either equation (pick the easy one)
6) Solve for other letter
7) Answer in (xy) format
bull 2x ndash y = 93x + 4y = -14
bull 4(2x ndash y = 9)
bull 8x ndash 4y = 36bull 3x + 4y = -14bull 11x = 22bull x = 2
bull 2(2) ndash y = 9bull -y = 5bull y = -5
bull (2 -5)
2x ndash y = 93x + 4y = -14
FAQs
Number next to a Letter (variable) means Multiply
4x
If x = -3
Then substitute amp multiply
4(-3) = -12
12a + -2b
If a = 5 b = -3
Then substitute amp multiply
12(5) + -2(-3) = 66
So do numbers next to ( ) and letters next to letters xy = xy
4(x) = 4x
(-2)(a) = -2aab = ab
4f g = 4fg
Addition and Subtraction are snobs They just combine with their own kind They form cliques
3x ndash 3a + 7 + 6x + 2a = 9x ndash a + 7
4xsup2 + 6x +9 ndash 15x + 3xsup2 + 10 = 7xsup2 - 9x + 19
Thatrsquos Just How They Do Thatrsquos How It Is
Deal With It
Multiplication and Division are party animals They will do it with anyone2x 4y = 8xy a bc = abc 12xy4x = 3y
Understanding Algebraic Culture is the Key to Success
23 = 2 divide 3
Numbers come with Signs (+ -)The sign is in front
Remember These are the same thing
5 ndash 3 = 25 + - 3 = 2
Because Subtraction is Adding the Opposite
If you are confused Circle the number amp the sign in front then do the math
2 ndash 56 + 7 ndash 8 ndash 10 = - 65
ABSOLUTE VALUE
The absolute value is always positive
The absolute value of 5 is 5The absolute value of -5 is 5
To solve drop the bars and make the inside number positive
Watch Out -6 = - 6 because the negative is lurking outside the bars
ALGEBRA OPPOSITES
Opposite of Multiplication is Division
25 4 = 100 100 divide 4 = 25
Positive Negative Numbers
The opposite of -5 is 5
The opposite of 6 is -6
Opposites add to zero
-4 + 4 = 0
Opposite of Addition is
Subtraction12 + 18 = 30
30 ndash 18 = 12
Opposite of Squaring is Square Rooting = 25 = 5
Adding Negative numbers - Think MONEY $$$$$$ You wonrsquot miss it
Algebra Truths
UGLY numbers work the
same as PRETTY Numbers
You canrsquot add FROGS amp
SMILEY FACES
LETTERS amp NUMBERS
work the same
Simplifying is cleaning up
your room put all the
FROGS together amp all the
SMILEY FACES together
then do the Math
No Equal Sign Simplify It Combine Like TermsSimplifying is like cleaning up your room put all the FROGS (Variables) together and all the SMILEY FACES (Numbers) together then do the Math Donrsquot Forget Numbers come with Signs
4x -5 +6x +21 -8x
4x-5
+6x+21
-8x
--------------------------------------------------------------------------------------------------------------------------------4x +6x -8x -5 +21
4x +6x -8x-5 +21
-------------------------------------------------------------------------------------------------------------------------------- 2x +16
2x +16
You canrsquot add FROGS amp SMILEY FACES so you are finished Good Job
WATCH YOUR SIGNS
Adding or Subtracting
If signs are the same add them and use the sign45 + 34 = 79 - 45 ndash 10 = -55
If signs are different subtract and use sign of larger number -18 + 8 = -10 60 ndash 20 = 40
Positive - dirt in the hole
Negative number -digging the hole
ndashPositive number dollars in your pocket Negative number dollars borrowed
Think temperature
Po
siti
ve i
s w
arm
ing
up
Neg
ative is coo
ling
off
Think moneyThink Holes
WATCH YOUR SIGNS
Multiplying or Dividing
If signs are the same answer is positive4 8 = 32 -63 -7 = -9
If signs are different answer is negative-6 7 = -42 -100 10 = -10
Distribution PrincipleMultiply everything in parenthesis by number next
to parenthesis FIRST THING
THINK BIG MOUTH SHOUTING ldquoDO MEDO ME FIRSTrdquo
EXAMPLE 3( 6x ndash 3)
3 (6x ndash 3) = 18x - 9
Distribution PrincipleMultiply everything in parenthesis by number next
to parenthesis FIRST THING
Get them off the busSo they can play football
EXAMPLE 3( 6x ndash 3)
3
18x - 9
6xndash 3
Example 2(10x ndash 3) = 6x + 2
Get the Teams off the Bus2(10x ndash 3) = 6x + 2
Line up the TeamsPenalty for off-sides ndash must change signs
Huddle up 14x = 14
Man on Man Defense 14x = 14 14 14
X = 1
Line of Scrimmage
20x ndash 6 = 6x + 8
20x ndash 6x = 8 + 6
Equation Solving
Think Football ndash Letters Vs Numbers
le rsaquo or lsaquo or geIf the equation has an Inequality sign follow the steps for solving equations with = signs
Play FootballLetters Vrs Numbers
Last Play of the Game
If you have to MULTIPLY or DIVIDE by a negative number Be sure to FLIP the Inequality
NOTICEUGLY NUMBERS WORK THE SAME AS PRETTY NUMBERS
Irsquom not afraid of
Fractions Have
Calculator Will
Calculate
UGLY Numbers Work the Same as PRETTY Numbers
If you can solve 2X + 10 = 40
Then you can solve 5x + 193 = 456
And you can solve
5x + ⅝ = ⅓
Play FootballLetters Vrs Numbers
Clue Words for writing equations from word problems
+
Word Clue
Plusadded to
the sum of increasing by
more than
Word Sentence
1 plus 56 is added to a numberThe sum of 5 and a numberA number is increased by 1015 is more than a number
Algebraic
1+56+x
5+x
x+10
x+15
Addition
Clue Words for writing equations from word problems
AlgebraicWord Clue
Minussubtracted fromthe difference ofdecreased byLessless than
Word Sentence
6 minus 57 subtracted from a numberThe difference of a number and 10A number is decreased by 205 less a number6 less than a number
6-5x-7
x-10
x-205-xx-6
Subtraction __
Clue Words for writing equations from word problems
Word Clue Word Sentence Algebraic
TimesProduct
DoubledTwiceOf (fractions and percents)
7 times a numberProduct of 8 and a numberA number doubledTwice a number12 a number55 of a number
7x = 7x8x
2x2x = 2x12x055x
Multiplication
Word Clue Word Sentence Algebraic
divide
Quotient
divided by
The quotient of a number and 710 divided by a number
xdivide7
10dividex
The first number written before the clue word will be the numerator
Clue Words for writing equations from word problems
Division
1 Consistent ndash one or many solutions2 Inconsistent ndash No solution
1 Independent ndash Only one solution2 Dependent ndash Has infinitely many
solutions
Slope Y ndash Int Graph Type of Systems
of Solutions
Same Same Same line ConsistentDependent
Infinitely many
Same Different Parallel Inconsistent 0
Different DifferentSame
Intersects ConsistentIndependent
1
Consistent and Inconsistent Systems
Slope ndash Intercept Form
y = mx + b
Slope- directions
RiseRun
Y Intercept ndash where to start
Itrsquos a line address
If the slope is a whole number put it on a stick m = 2 slope is 21
To Graph
y = 2X + 1 Starts at 1
Riserun = 21
Directions are up 2 over 1
Example 1 Example 2y = -3X+ 0
y = -3X
Starts at 0
riserun = 3-1
Directions are up 3 over -1
Thanks to httpwwwmathsisfuncomequation_of_linehtml
Linear Equations Standard Form ax + by = cSolving for y Just 3 easy stepsGreenisms Math Terms
1Move x term(Change sides Change signs) AddSubtract x term 2Give x side a hug Parenthesis ( )3Divide by number next to the y Coefficient
Example Solve for Y2x ndash 7y = 12Just 3 easy steps1 -7y = 12 ndash 2x (Move x term (Change sides Change signs)2 -7y = (12-2x) (Give it a hug)3 y = (12-2x) -7 (Divide by number next to the y)
Now you are ready to enter it into the calculator and graph it
WATCH YOUR SIGNS
Example Solve for Y2x ndash 7y = 12
Just 3 easy steps1 -7y = 12 ndash 2x X is offside Penalty change signs2 -7y = (12-2x) Huddle up ( )3 y = (12-2x) -7 Man on man defense
WATCH YOUR SIGNS
Linear Equations Standard Form ax + by = cSolving for y Itrsquos a Football Game
Y VS Everybody Else
Follow football rules
Play FootballY vs everybody else
Now you are ready to enter it into the calculator and graph it
l lines
Same Slope
Slopes are Negative Reciprocal
(Flip amp Change Sign)
PARALLEL LINES
y2 ndash y1
x2 ndash x1
or
y = mx + b
slope
Find Equation of the Line y = mx + b
I need slope (m) amp the y-intercept
(b)
MY ANSWER
y = x +
To find m ndash Solve the equation for y and use mor use the y2 ndash y1
x2 ndash x1 formula
To find b - Plug x y and m into the line equation and solve for b
(Just follow the Rules)
Base xsup2Exponent
Like Bases Exponent Example
Multiply Add
Power up Multiply
Divide Subtract s
Add No change 3asup2 + 5asup2 = 8asup2 asup2 + = asup2 +
Subtract No change 10asup2 - 4asup2 = 6asup2
5a
4
15 9 3a g y
6 4 5
3 7 4a x ka x k
5a
5a
5a
Exponents
=
(xsup2asup3g)(xasup2gsup3) = (xsup3 g )
( gsup3 y) sup3 =
asup3k xsup3
Negative ExponentsJust switch places and make exponent positive
5x5
1x
Example Switch Simplify
5 4 3
2 2 4x g ka x k
4 2
2 5 4 3g x
a x k k
4
2 3 7g
a x k
=
Xsup2
X X∙ = Xsup2 X
X times X is X squared
X
QuadraticsMultiplying Binomials ndash Draw the Face
(x + 8) (x ndash 6)
Multiply Watch your Signs
xsup2 +8x -6x -48
Simplify (Combine Like Terms eat the buggers)xsup2 + 2x -48
Check This First
The Math ldquoFrdquo Word ldquoFactoringrdquo
1 Is there a common factor (number or letter)
NO
Proceed to question 2
Yes
Proceed with CGF
4xsup3 2 2 x x x
4x ( 2xy + xsup2 -3 )
Example
8xsup2y + 4xsup3 - 12x
Factor each term8xsup2y 2 2 2 x x y 12x 2 2 3 x
Circle common terms
8xsup2y 2 2 2 x x y 4xsup3 2 2 x x x 12x 2 2 3 x
Multiply circled numbersThatrsquos your Common Factor
Multiply leftovers put in ( )
2 Perfect squares on end amp 3 terms
xsup2 - 10x + 25
YES SPLIT IT NICE
NO proceed to question 3
xsup2 - 10x + 25Put out baggies to hold the answer (parenthesis)
( x - 5 ) ( x - 5 )
Split the first term nice
Place in baggies
Sign between is same as middle term
Split the second term nice
Place in baggies
ANSWER
(x -5)sup2
3 Perfect squares on end amp 2 terms
YesSame as question 2Except the signsare +-
Example
xsup2 - 64
NO proceed to question 4
( x - 8 ) ( x + 8)
Answer looks like this
4 No perfect squares on end 3 terms amp starts with xsup2
NO proceed to question 5
YES its quadratics in the morning
xsup2 - 10x + 24MA
Multiply to +24 Add to -10
38 11
-3 -8 -11
-2 -12 -14
-6-4 -10 Found it
Put in the parenthesis
(x-6) (x-4)
All Done
2 Search and Seizure (quadratics in the morning)(See question 4)
1 Steal the ldquoardquo and give it to the last term (Multiply)1
5 Is there a number in front of the xsup2 amp does it have 3 terms
Yes Jail BreakNO
Then it is Prime
(canrsquot factor)
Example 2xsup2 + 7x + 3
xsup2 + 7x + 6 (23)
( x + 6 ) ( x + 1 )
3 Arrested and Caught - Divide last terms by ldquoardquo
( x + 62 ) ( x + 12 )
4 Beat it Down - (reduce fractions)
( x + 3 ) ( x + 12 )
5 Parole (kick denominator to the front)(x + 3 ) ( 2 x + 1 )
All Done
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Check This First
Is there a common factor (number or letter)
Greatest Common Factor (GCF)
1 Factor each term2 Circle common terms3 Multiply common terms4 Write it down5 Put out baggie for leftover6 Multiply leftovers for each
term7 Put in baggie ( )
Example
8xsup2y + 4xsup3 - 12x
4x ( 2xy + xsup2 - 3 )
Is there a Square on each End
Perfect Squares
1 Put out Parenthesis2 Split FIRST and LAST numbers nice3 Put in Signs
3 Different Kinds
A xsup2 - 16x + 64
(xndash 8) (x ndash 8)
B xsup2 + 18x + 81
(x + 9) (x + 9)
C xsup2 - 36
(x + 6) (x ndash 6)
Is there a number in front of the xsup2 and does it have 3 terms
Jail Break
1Steal the ldquoardquo and give it to the last term (multiply)2Search and Seizure (Quad in AM)3Arrested and Caught (divide by ldquoardquo)4Beat Down (reduce fractions)5Parole (kick denominator to the front)6Check it out (FACE it)
Example
2xsup2 + 7x + 3
1xsup2 + 7x + 62(x + 6) (x + 1)3(x + 62) (x + 12)4(x + 3) (x +12)5(x + 3) (2x + 1)
No perfect squares and 3 terms and starts with xsup2
Quadratics in the Morning (AM)
1 Make a factor tree2 Multiply to last number add
to middle number3 Put out baggies (parenthesis)4 Split first term nice5 Drop in factors from tree
Example
xsup2 + 12x + 32
(x + 8) (x + 4)
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Solve with Graphing Calculator
1) Solve each equation for y (3 easy steps)
2) Use y= button and enter each equation
3) Use graph to eyeball answerOr
4) Use to find where y1 and y2 are equal
5) Be sure answer is in (x y) form
Y=
Systems of EquationsTo Solve by Graphing
1) Make an x y Chart
2) Select any x solve for y x 2x ndash 4 y 0 2(0) ndash 4 -4 1 2(1) ndash 4 -2
3) Then graph the two points4) Do for both equations5) The answer is where they
cross6) Be sure answer is in (x y)
form
2nd TABLE
bull y + 2x = 9
bull y = 9 ndash 2xbull 3(9 - 2x) ndash 2x = 11
bull 27-6x ndash 2x = 11bull 27 ndash 8x = 11bull - 8x = -16bull x = 2bull y = 9 ndash 2 (2)bull y = 5bull (2 5)
Systems of EquationsSolve by Substitution
(box amp shove)
1) Solve one equation for x or y (change sides change signs)
2) Box it3) Rewrite other equation
and shove box in 4) Solve for surviving letter
1) Distribute2) Combine like terms3) Solve
4) Send it back to box5) Solve for other letter6) Answer in (xy) form
3y -2x = 11y + 2x = 9
Systems of EquationsSolve by Elimination
1) Look for opposite signs
2) Multiply to create opposites
3) Add old and new equation together
4) Solve for surviving letter
5) Plug back into either equation (pick the easy one)
6) Solve for other letter
7) Answer in (xy) format
bull 2x ndash y = 93x + 4y = -14
bull 4(2x ndash y = 9)
bull 8x ndash 4y = 36bull 3x + 4y = -14bull 11x = 22bull x = 2
bull 2(2) ndash y = 9bull -y = 5bull y = -5
bull (2 -5)
2x ndash y = 93x + 4y = -14
Number next to a Letter (variable) means Multiply
4x
If x = -3
Then substitute amp multiply
4(-3) = -12
12a + -2b
If a = 5 b = -3
Then substitute amp multiply
12(5) + -2(-3) = 66
So do numbers next to ( ) and letters next to letters xy = xy
4(x) = 4x
(-2)(a) = -2aab = ab
4f g = 4fg
Addition and Subtraction are snobs They just combine with their own kind They form cliques
3x ndash 3a + 7 + 6x + 2a = 9x ndash a + 7
4xsup2 + 6x +9 ndash 15x + 3xsup2 + 10 = 7xsup2 - 9x + 19
Thatrsquos Just How They Do Thatrsquos How It Is
Deal With It
Multiplication and Division are party animals They will do it with anyone2x 4y = 8xy a bc = abc 12xy4x = 3y
Understanding Algebraic Culture is the Key to Success
23 = 2 divide 3
Numbers come with Signs (+ -)The sign is in front
Remember These are the same thing
5 ndash 3 = 25 + - 3 = 2
Because Subtraction is Adding the Opposite
If you are confused Circle the number amp the sign in front then do the math
2 ndash 56 + 7 ndash 8 ndash 10 = - 65
ABSOLUTE VALUE
The absolute value is always positive
The absolute value of 5 is 5The absolute value of -5 is 5
To solve drop the bars and make the inside number positive
Watch Out -6 = - 6 because the negative is lurking outside the bars
ALGEBRA OPPOSITES
Opposite of Multiplication is Division
25 4 = 100 100 divide 4 = 25
Positive Negative Numbers
The opposite of -5 is 5
The opposite of 6 is -6
Opposites add to zero
-4 + 4 = 0
Opposite of Addition is
Subtraction12 + 18 = 30
30 ndash 18 = 12
Opposite of Squaring is Square Rooting = 25 = 5
Adding Negative numbers - Think MONEY $$$$$$ You wonrsquot miss it
Algebra Truths
UGLY numbers work the
same as PRETTY Numbers
You canrsquot add FROGS amp
SMILEY FACES
LETTERS amp NUMBERS
work the same
Simplifying is cleaning up
your room put all the
FROGS together amp all the
SMILEY FACES together
then do the Math
No Equal Sign Simplify It Combine Like TermsSimplifying is like cleaning up your room put all the FROGS (Variables) together and all the SMILEY FACES (Numbers) together then do the Math Donrsquot Forget Numbers come with Signs
4x -5 +6x +21 -8x
4x-5
+6x+21
-8x
--------------------------------------------------------------------------------------------------------------------------------4x +6x -8x -5 +21
4x +6x -8x-5 +21
-------------------------------------------------------------------------------------------------------------------------------- 2x +16
2x +16
You canrsquot add FROGS amp SMILEY FACES so you are finished Good Job
WATCH YOUR SIGNS
Adding or Subtracting
If signs are the same add them and use the sign45 + 34 = 79 - 45 ndash 10 = -55
If signs are different subtract and use sign of larger number -18 + 8 = -10 60 ndash 20 = 40
Positive - dirt in the hole
Negative number -digging the hole
ndashPositive number dollars in your pocket Negative number dollars borrowed
Think temperature
Po
siti
ve i
s w
arm
ing
up
Neg
ative is coo
ling
off
Think moneyThink Holes
WATCH YOUR SIGNS
Multiplying or Dividing
If signs are the same answer is positive4 8 = 32 -63 -7 = -9
If signs are different answer is negative-6 7 = -42 -100 10 = -10
Distribution PrincipleMultiply everything in parenthesis by number next
to parenthesis FIRST THING
THINK BIG MOUTH SHOUTING ldquoDO MEDO ME FIRSTrdquo
EXAMPLE 3( 6x ndash 3)
3 (6x ndash 3) = 18x - 9
Distribution PrincipleMultiply everything in parenthesis by number next
to parenthesis FIRST THING
Get them off the busSo they can play football
EXAMPLE 3( 6x ndash 3)
3
18x - 9
6xndash 3
Example 2(10x ndash 3) = 6x + 2
Get the Teams off the Bus2(10x ndash 3) = 6x + 2
Line up the TeamsPenalty for off-sides ndash must change signs
Huddle up 14x = 14
Man on Man Defense 14x = 14 14 14
X = 1
Line of Scrimmage
20x ndash 6 = 6x + 8
20x ndash 6x = 8 + 6
Equation Solving
Think Football ndash Letters Vs Numbers
le rsaquo or lsaquo or geIf the equation has an Inequality sign follow the steps for solving equations with = signs
Play FootballLetters Vrs Numbers
Last Play of the Game
If you have to MULTIPLY or DIVIDE by a negative number Be sure to FLIP the Inequality
NOTICEUGLY NUMBERS WORK THE SAME AS PRETTY NUMBERS
Irsquom not afraid of
Fractions Have
Calculator Will
Calculate
UGLY Numbers Work the Same as PRETTY Numbers
If you can solve 2X + 10 = 40
Then you can solve 5x + 193 = 456
And you can solve
5x + ⅝ = ⅓
Play FootballLetters Vrs Numbers
Clue Words for writing equations from word problems
+
Word Clue
Plusadded to
the sum of increasing by
more than
Word Sentence
1 plus 56 is added to a numberThe sum of 5 and a numberA number is increased by 1015 is more than a number
Algebraic
1+56+x
5+x
x+10
x+15
Addition
Clue Words for writing equations from word problems
AlgebraicWord Clue
Minussubtracted fromthe difference ofdecreased byLessless than
Word Sentence
6 minus 57 subtracted from a numberThe difference of a number and 10A number is decreased by 205 less a number6 less than a number
6-5x-7
x-10
x-205-xx-6
Subtraction __
Clue Words for writing equations from word problems
Word Clue Word Sentence Algebraic
TimesProduct
DoubledTwiceOf (fractions and percents)
7 times a numberProduct of 8 and a numberA number doubledTwice a number12 a number55 of a number
7x = 7x8x
2x2x = 2x12x055x
Multiplication
Word Clue Word Sentence Algebraic
divide
Quotient
divided by
The quotient of a number and 710 divided by a number
xdivide7
10dividex
The first number written before the clue word will be the numerator
Clue Words for writing equations from word problems
Division
1 Consistent ndash one or many solutions2 Inconsistent ndash No solution
1 Independent ndash Only one solution2 Dependent ndash Has infinitely many
solutions
Slope Y ndash Int Graph Type of Systems
of Solutions
Same Same Same line ConsistentDependent
Infinitely many
Same Different Parallel Inconsistent 0
Different DifferentSame
Intersects ConsistentIndependent
1
Consistent and Inconsistent Systems
Slope ndash Intercept Form
y = mx + b
Slope- directions
RiseRun
Y Intercept ndash where to start
Itrsquos a line address
If the slope is a whole number put it on a stick m = 2 slope is 21
To Graph
y = 2X + 1 Starts at 1
Riserun = 21
Directions are up 2 over 1
Example 1 Example 2y = -3X+ 0
y = -3X
Starts at 0
riserun = 3-1
Directions are up 3 over -1
Thanks to httpwwwmathsisfuncomequation_of_linehtml
Linear Equations Standard Form ax + by = cSolving for y Just 3 easy stepsGreenisms Math Terms
1Move x term(Change sides Change signs) AddSubtract x term 2Give x side a hug Parenthesis ( )3Divide by number next to the y Coefficient
Example Solve for Y2x ndash 7y = 12Just 3 easy steps1 -7y = 12 ndash 2x (Move x term (Change sides Change signs)2 -7y = (12-2x) (Give it a hug)3 y = (12-2x) -7 (Divide by number next to the y)
Now you are ready to enter it into the calculator and graph it
WATCH YOUR SIGNS
Example Solve for Y2x ndash 7y = 12
Just 3 easy steps1 -7y = 12 ndash 2x X is offside Penalty change signs2 -7y = (12-2x) Huddle up ( )3 y = (12-2x) -7 Man on man defense
WATCH YOUR SIGNS
Linear Equations Standard Form ax + by = cSolving for y Itrsquos a Football Game
Y VS Everybody Else
Follow football rules
Play FootballY vs everybody else
Now you are ready to enter it into the calculator and graph it
l lines
Same Slope
Slopes are Negative Reciprocal
(Flip amp Change Sign)
PARALLEL LINES
y2 ndash y1
x2 ndash x1
or
y = mx + b
slope
Find Equation of the Line y = mx + b
I need slope (m) amp the y-intercept
(b)
MY ANSWER
y = x +
To find m ndash Solve the equation for y and use mor use the y2 ndash y1
x2 ndash x1 formula
To find b - Plug x y and m into the line equation and solve for b
(Just follow the Rules)
Base xsup2Exponent
Like Bases Exponent Example
Multiply Add
Power up Multiply
Divide Subtract s
Add No change 3asup2 + 5asup2 = 8asup2 asup2 + = asup2 +
Subtract No change 10asup2 - 4asup2 = 6asup2
5a
4
15 9 3a g y
6 4 5
3 7 4a x ka x k
5a
5a
5a
Exponents
=
(xsup2asup3g)(xasup2gsup3) = (xsup3 g )
( gsup3 y) sup3 =
asup3k xsup3
Negative ExponentsJust switch places and make exponent positive
5x5
1x
Example Switch Simplify
5 4 3
2 2 4x g ka x k
4 2
2 5 4 3g x
a x k k
4
2 3 7g
a x k
=
Xsup2
X X∙ = Xsup2 X
X times X is X squared
X
QuadraticsMultiplying Binomials ndash Draw the Face
(x + 8) (x ndash 6)
Multiply Watch your Signs
xsup2 +8x -6x -48
Simplify (Combine Like Terms eat the buggers)xsup2 + 2x -48
Check This First
The Math ldquoFrdquo Word ldquoFactoringrdquo
1 Is there a common factor (number or letter)
NO
Proceed to question 2
Yes
Proceed with CGF
4xsup3 2 2 x x x
4x ( 2xy + xsup2 -3 )
Example
8xsup2y + 4xsup3 - 12x
Factor each term8xsup2y 2 2 2 x x y 12x 2 2 3 x
Circle common terms
8xsup2y 2 2 2 x x y 4xsup3 2 2 x x x 12x 2 2 3 x
Multiply circled numbersThatrsquos your Common Factor
Multiply leftovers put in ( )
2 Perfect squares on end amp 3 terms
xsup2 - 10x + 25
YES SPLIT IT NICE
NO proceed to question 3
xsup2 - 10x + 25Put out baggies to hold the answer (parenthesis)
( x - 5 ) ( x - 5 )
Split the first term nice
Place in baggies
Sign between is same as middle term
Split the second term nice
Place in baggies
ANSWER
(x -5)sup2
3 Perfect squares on end amp 2 terms
YesSame as question 2Except the signsare +-
Example
xsup2 - 64
NO proceed to question 4
( x - 8 ) ( x + 8)
Answer looks like this
4 No perfect squares on end 3 terms amp starts with xsup2
NO proceed to question 5
YES its quadratics in the morning
xsup2 - 10x + 24MA
Multiply to +24 Add to -10
38 11
-3 -8 -11
-2 -12 -14
-6-4 -10 Found it
Put in the parenthesis
(x-6) (x-4)
All Done
2 Search and Seizure (quadratics in the morning)(See question 4)
1 Steal the ldquoardquo and give it to the last term (Multiply)1
5 Is there a number in front of the xsup2 amp does it have 3 terms
Yes Jail BreakNO
Then it is Prime
(canrsquot factor)
Example 2xsup2 + 7x + 3
xsup2 + 7x + 6 (23)
( x + 6 ) ( x + 1 )
3 Arrested and Caught - Divide last terms by ldquoardquo
( x + 62 ) ( x + 12 )
4 Beat it Down - (reduce fractions)
( x + 3 ) ( x + 12 )
5 Parole (kick denominator to the front)(x + 3 ) ( 2 x + 1 )
All Done
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Check This First
Is there a common factor (number or letter)
Greatest Common Factor (GCF)
1 Factor each term2 Circle common terms3 Multiply common terms4 Write it down5 Put out baggie for leftover6 Multiply leftovers for each
term7 Put in baggie ( )
Example
8xsup2y + 4xsup3 - 12x
4x ( 2xy + xsup2 - 3 )
Is there a Square on each End
Perfect Squares
1 Put out Parenthesis2 Split FIRST and LAST numbers nice3 Put in Signs
3 Different Kinds
A xsup2 - 16x + 64
(xndash 8) (x ndash 8)
B xsup2 + 18x + 81
(x + 9) (x + 9)
C xsup2 - 36
(x + 6) (x ndash 6)
Is there a number in front of the xsup2 and does it have 3 terms
Jail Break
1Steal the ldquoardquo and give it to the last term (multiply)2Search and Seizure (Quad in AM)3Arrested and Caught (divide by ldquoardquo)4Beat Down (reduce fractions)5Parole (kick denominator to the front)6Check it out (FACE it)
Example
2xsup2 + 7x + 3
1xsup2 + 7x + 62(x + 6) (x + 1)3(x + 62) (x + 12)4(x + 3) (x +12)5(x + 3) (2x + 1)
No perfect squares and 3 terms and starts with xsup2
Quadratics in the Morning (AM)
1 Make a factor tree2 Multiply to last number add
to middle number3 Put out baggies (parenthesis)4 Split first term nice5 Drop in factors from tree
Example
xsup2 + 12x + 32
(x + 8) (x + 4)
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Solve with Graphing Calculator
1) Solve each equation for y (3 easy steps)
2) Use y= button and enter each equation
3) Use graph to eyeball answerOr
4) Use to find where y1 and y2 are equal
5) Be sure answer is in (x y) form
Y=
Systems of EquationsTo Solve by Graphing
1) Make an x y Chart
2) Select any x solve for y x 2x ndash 4 y 0 2(0) ndash 4 -4 1 2(1) ndash 4 -2
3) Then graph the two points4) Do for both equations5) The answer is where they
cross6) Be sure answer is in (x y)
form
2nd TABLE
bull y + 2x = 9
bull y = 9 ndash 2xbull 3(9 - 2x) ndash 2x = 11
bull 27-6x ndash 2x = 11bull 27 ndash 8x = 11bull - 8x = -16bull x = 2bull y = 9 ndash 2 (2)bull y = 5bull (2 5)
Systems of EquationsSolve by Substitution
(box amp shove)
1) Solve one equation for x or y (change sides change signs)
2) Box it3) Rewrite other equation
and shove box in 4) Solve for surviving letter
1) Distribute2) Combine like terms3) Solve
4) Send it back to box5) Solve for other letter6) Answer in (xy) form
3y -2x = 11y + 2x = 9
Systems of EquationsSolve by Elimination
1) Look for opposite signs
2) Multiply to create opposites
3) Add old and new equation together
4) Solve for surviving letter
5) Plug back into either equation (pick the easy one)
6) Solve for other letter
7) Answer in (xy) format
bull 2x ndash y = 93x + 4y = -14
bull 4(2x ndash y = 9)
bull 8x ndash 4y = 36bull 3x + 4y = -14bull 11x = 22bull x = 2
bull 2(2) ndash y = 9bull -y = 5bull y = -5
bull (2 -5)
2x ndash y = 93x + 4y = -14
Addition and Subtraction are snobs They just combine with their own kind They form cliques
3x ndash 3a + 7 + 6x + 2a = 9x ndash a + 7
4xsup2 + 6x +9 ndash 15x + 3xsup2 + 10 = 7xsup2 - 9x + 19
Thatrsquos Just How They Do Thatrsquos How It Is
Deal With It
Multiplication and Division are party animals They will do it with anyone2x 4y = 8xy a bc = abc 12xy4x = 3y
Understanding Algebraic Culture is the Key to Success
23 = 2 divide 3
Numbers come with Signs (+ -)The sign is in front
Remember These are the same thing
5 ndash 3 = 25 + - 3 = 2
Because Subtraction is Adding the Opposite
If you are confused Circle the number amp the sign in front then do the math
2 ndash 56 + 7 ndash 8 ndash 10 = - 65
ABSOLUTE VALUE
The absolute value is always positive
The absolute value of 5 is 5The absolute value of -5 is 5
To solve drop the bars and make the inside number positive
Watch Out -6 = - 6 because the negative is lurking outside the bars
ALGEBRA OPPOSITES
Opposite of Multiplication is Division
25 4 = 100 100 divide 4 = 25
Positive Negative Numbers
The opposite of -5 is 5
The opposite of 6 is -6
Opposites add to zero
-4 + 4 = 0
Opposite of Addition is
Subtraction12 + 18 = 30
30 ndash 18 = 12
Opposite of Squaring is Square Rooting = 25 = 5
Adding Negative numbers - Think MONEY $$$$$$ You wonrsquot miss it
Algebra Truths
UGLY numbers work the
same as PRETTY Numbers
You canrsquot add FROGS amp
SMILEY FACES
LETTERS amp NUMBERS
work the same
Simplifying is cleaning up
your room put all the
FROGS together amp all the
SMILEY FACES together
then do the Math
No Equal Sign Simplify It Combine Like TermsSimplifying is like cleaning up your room put all the FROGS (Variables) together and all the SMILEY FACES (Numbers) together then do the Math Donrsquot Forget Numbers come with Signs
4x -5 +6x +21 -8x
4x-5
+6x+21
-8x
--------------------------------------------------------------------------------------------------------------------------------4x +6x -8x -5 +21
4x +6x -8x-5 +21
-------------------------------------------------------------------------------------------------------------------------------- 2x +16
2x +16
You canrsquot add FROGS amp SMILEY FACES so you are finished Good Job
WATCH YOUR SIGNS
Adding or Subtracting
If signs are the same add them and use the sign45 + 34 = 79 - 45 ndash 10 = -55
If signs are different subtract and use sign of larger number -18 + 8 = -10 60 ndash 20 = 40
Positive - dirt in the hole
Negative number -digging the hole
ndashPositive number dollars in your pocket Negative number dollars borrowed
Think temperature
Po
siti
ve i
s w
arm
ing
up
Neg
ative is coo
ling
off
Think moneyThink Holes
WATCH YOUR SIGNS
Multiplying or Dividing
If signs are the same answer is positive4 8 = 32 -63 -7 = -9
If signs are different answer is negative-6 7 = -42 -100 10 = -10
Distribution PrincipleMultiply everything in parenthesis by number next
to parenthesis FIRST THING
THINK BIG MOUTH SHOUTING ldquoDO MEDO ME FIRSTrdquo
EXAMPLE 3( 6x ndash 3)
3 (6x ndash 3) = 18x - 9
Distribution PrincipleMultiply everything in parenthesis by number next
to parenthesis FIRST THING
Get them off the busSo they can play football
EXAMPLE 3( 6x ndash 3)
3
18x - 9
6xndash 3
Example 2(10x ndash 3) = 6x + 2
Get the Teams off the Bus2(10x ndash 3) = 6x + 2
Line up the TeamsPenalty for off-sides ndash must change signs
Huddle up 14x = 14
Man on Man Defense 14x = 14 14 14
X = 1
Line of Scrimmage
20x ndash 6 = 6x + 8
20x ndash 6x = 8 + 6
Equation Solving
Think Football ndash Letters Vs Numbers
le rsaquo or lsaquo or geIf the equation has an Inequality sign follow the steps for solving equations with = signs
Play FootballLetters Vrs Numbers
Last Play of the Game
If you have to MULTIPLY or DIVIDE by a negative number Be sure to FLIP the Inequality
NOTICEUGLY NUMBERS WORK THE SAME AS PRETTY NUMBERS
Irsquom not afraid of
Fractions Have
Calculator Will
Calculate
UGLY Numbers Work the Same as PRETTY Numbers
If you can solve 2X + 10 = 40
Then you can solve 5x + 193 = 456
And you can solve
5x + ⅝ = ⅓
Play FootballLetters Vrs Numbers
Clue Words for writing equations from word problems
+
Word Clue
Plusadded to
the sum of increasing by
more than
Word Sentence
1 plus 56 is added to a numberThe sum of 5 and a numberA number is increased by 1015 is more than a number
Algebraic
1+56+x
5+x
x+10
x+15
Addition
Clue Words for writing equations from word problems
AlgebraicWord Clue
Minussubtracted fromthe difference ofdecreased byLessless than
Word Sentence
6 minus 57 subtracted from a numberThe difference of a number and 10A number is decreased by 205 less a number6 less than a number
6-5x-7
x-10
x-205-xx-6
Subtraction __
Clue Words for writing equations from word problems
Word Clue Word Sentence Algebraic
TimesProduct
DoubledTwiceOf (fractions and percents)
7 times a numberProduct of 8 and a numberA number doubledTwice a number12 a number55 of a number
7x = 7x8x
2x2x = 2x12x055x
Multiplication
Word Clue Word Sentence Algebraic
divide
Quotient
divided by
The quotient of a number and 710 divided by a number
xdivide7
10dividex
The first number written before the clue word will be the numerator
Clue Words for writing equations from word problems
Division
1 Consistent ndash one or many solutions2 Inconsistent ndash No solution
1 Independent ndash Only one solution2 Dependent ndash Has infinitely many
solutions
Slope Y ndash Int Graph Type of Systems
of Solutions
Same Same Same line ConsistentDependent
Infinitely many
Same Different Parallel Inconsistent 0
Different DifferentSame
Intersects ConsistentIndependent
1
Consistent and Inconsistent Systems
Slope ndash Intercept Form
y = mx + b
Slope- directions
RiseRun
Y Intercept ndash where to start
Itrsquos a line address
If the slope is a whole number put it on a stick m = 2 slope is 21
To Graph
y = 2X + 1 Starts at 1
Riserun = 21
Directions are up 2 over 1
Example 1 Example 2y = -3X+ 0
y = -3X
Starts at 0
riserun = 3-1
Directions are up 3 over -1
Thanks to httpwwwmathsisfuncomequation_of_linehtml
Linear Equations Standard Form ax + by = cSolving for y Just 3 easy stepsGreenisms Math Terms
1Move x term(Change sides Change signs) AddSubtract x term 2Give x side a hug Parenthesis ( )3Divide by number next to the y Coefficient
Example Solve for Y2x ndash 7y = 12Just 3 easy steps1 -7y = 12 ndash 2x (Move x term (Change sides Change signs)2 -7y = (12-2x) (Give it a hug)3 y = (12-2x) -7 (Divide by number next to the y)
Now you are ready to enter it into the calculator and graph it
WATCH YOUR SIGNS
Example Solve for Y2x ndash 7y = 12
Just 3 easy steps1 -7y = 12 ndash 2x X is offside Penalty change signs2 -7y = (12-2x) Huddle up ( )3 y = (12-2x) -7 Man on man defense
WATCH YOUR SIGNS
Linear Equations Standard Form ax + by = cSolving for y Itrsquos a Football Game
Y VS Everybody Else
Follow football rules
Play FootballY vs everybody else
Now you are ready to enter it into the calculator and graph it
l lines
Same Slope
Slopes are Negative Reciprocal
(Flip amp Change Sign)
PARALLEL LINES
y2 ndash y1
x2 ndash x1
or
y = mx + b
slope
Find Equation of the Line y = mx + b
I need slope (m) amp the y-intercept
(b)
MY ANSWER
y = x +
To find m ndash Solve the equation for y and use mor use the y2 ndash y1
x2 ndash x1 formula
To find b - Plug x y and m into the line equation and solve for b
(Just follow the Rules)
Base xsup2Exponent
Like Bases Exponent Example
Multiply Add
Power up Multiply
Divide Subtract s
Add No change 3asup2 + 5asup2 = 8asup2 asup2 + = asup2 +
Subtract No change 10asup2 - 4asup2 = 6asup2
5a
4
15 9 3a g y
6 4 5
3 7 4a x ka x k
5a
5a
5a
Exponents
=
(xsup2asup3g)(xasup2gsup3) = (xsup3 g )
( gsup3 y) sup3 =
asup3k xsup3
Negative ExponentsJust switch places and make exponent positive
5x5
1x
Example Switch Simplify
5 4 3
2 2 4x g ka x k
4 2
2 5 4 3g x
a x k k
4
2 3 7g
a x k
=
Xsup2
X X∙ = Xsup2 X
X times X is X squared
X
QuadraticsMultiplying Binomials ndash Draw the Face
(x + 8) (x ndash 6)
Multiply Watch your Signs
xsup2 +8x -6x -48
Simplify (Combine Like Terms eat the buggers)xsup2 + 2x -48
Check This First
The Math ldquoFrdquo Word ldquoFactoringrdquo
1 Is there a common factor (number or letter)
NO
Proceed to question 2
Yes
Proceed with CGF
4xsup3 2 2 x x x
4x ( 2xy + xsup2 -3 )
Example
8xsup2y + 4xsup3 - 12x
Factor each term8xsup2y 2 2 2 x x y 12x 2 2 3 x
Circle common terms
8xsup2y 2 2 2 x x y 4xsup3 2 2 x x x 12x 2 2 3 x
Multiply circled numbersThatrsquos your Common Factor
Multiply leftovers put in ( )
2 Perfect squares on end amp 3 terms
xsup2 - 10x + 25
YES SPLIT IT NICE
NO proceed to question 3
xsup2 - 10x + 25Put out baggies to hold the answer (parenthesis)
( x - 5 ) ( x - 5 )
Split the first term nice
Place in baggies
Sign between is same as middle term
Split the second term nice
Place in baggies
ANSWER
(x -5)sup2
3 Perfect squares on end amp 2 terms
YesSame as question 2Except the signsare +-
Example
xsup2 - 64
NO proceed to question 4
( x - 8 ) ( x + 8)
Answer looks like this
4 No perfect squares on end 3 terms amp starts with xsup2
NO proceed to question 5
YES its quadratics in the morning
xsup2 - 10x + 24MA
Multiply to +24 Add to -10
38 11
-3 -8 -11
-2 -12 -14
-6-4 -10 Found it
Put in the parenthesis
(x-6) (x-4)
All Done
2 Search and Seizure (quadratics in the morning)(See question 4)
1 Steal the ldquoardquo and give it to the last term (Multiply)1
5 Is there a number in front of the xsup2 amp does it have 3 terms
Yes Jail BreakNO
Then it is Prime
(canrsquot factor)
Example 2xsup2 + 7x + 3
xsup2 + 7x + 6 (23)
( x + 6 ) ( x + 1 )
3 Arrested and Caught - Divide last terms by ldquoardquo
( x + 62 ) ( x + 12 )
4 Beat it Down - (reduce fractions)
( x + 3 ) ( x + 12 )
5 Parole (kick denominator to the front)(x + 3 ) ( 2 x + 1 )
All Done
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Check This First
Is there a common factor (number or letter)
Greatest Common Factor (GCF)
1 Factor each term2 Circle common terms3 Multiply common terms4 Write it down5 Put out baggie for leftover6 Multiply leftovers for each
term7 Put in baggie ( )
Example
8xsup2y + 4xsup3 - 12x
4x ( 2xy + xsup2 - 3 )
Is there a Square on each End
Perfect Squares
1 Put out Parenthesis2 Split FIRST and LAST numbers nice3 Put in Signs
3 Different Kinds
A xsup2 - 16x + 64
(xndash 8) (x ndash 8)
B xsup2 + 18x + 81
(x + 9) (x + 9)
C xsup2 - 36
(x + 6) (x ndash 6)
Is there a number in front of the xsup2 and does it have 3 terms
Jail Break
1Steal the ldquoardquo and give it to the last term (multiply)2Search and Seizure (Quad in AM)3Arrested and Caught (divide by ldquoardquo)4Beat Down (reduce fractions)5Parole (kick denominator to the front)6Check it out (FACE it)
Example
2xsup2 + 7x + 3
1xsup2 + 7x + 62(x + 6) (x + 1)3(x + 62) (x + 12)4(x + 3) (x +12)5(x + 3) (2x + 1)
No perfect squares and 3 terms and starts with xsup2
Quadratics in the Morning (AM)
1 Make a factor tree2 Multiply to last number add
to middle number3 Put out baggies (parenthesis)4 Split first term nice5 Drop in factors from tree
Example
xsup2 + 12x + 32
(x + 8) (x + 4)
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Solve with Graphing Calculator
1) Solve each equation for y (3 easy steps)
2) Use y= button and enter each equation
3) Use graph to eyeball answerOr
4) Use to find where y1 and y2 are equal
5) Be sure answer is in (x y) form
Y=
Systems of EquationsTo Solve by Graphing
1) Make an x y Chart
2) Select any x solve for y x 2x ndash 4 y 0 2(0) ndash 4 -4 1 2(1) ndash 4 -2
3) Then graph the two points4) Do for both equations5) The answer is where they
cross6) Be sure answer is in (x y)
form
2nd TABLE
bull y + 2x = 9
bull y = 9 ndash 2xbull 3(9 - 2x) ndash 2x = 11
bull 27-6x ndash 2x = 11bull 27 ndash 8x = 11bull - 8x = -16bull x = 2bull y = 9 ndash 2 (2)bull y = 5bull (2 5)
Systems of EquationsSolve by Substitution
(box amp shove)
1) Solve one equation for x or y (change sides change signs)
2) Box it3) Rewrite other equation
and shove box in 4) Solve for surviving letter
1) Distribute2) Combine like terms3) Solve
4) Send it back to box5) Solve for other letter6) Answer in (xy) form
3y -2x = 11y + 2x = 9
Systems of EquationsSolve by Elimination
1) Look for opposite signs
2) Multiply to create opposites
3) Add old and new equation together
4) Solve for surviving letter
5) Plug back into either equation (pick the easy one)
6) Solve for other letter
7) Answer in (xy) format
bull 2x ndash y = 93x + 4y = -14
bull 4(2x ndash y = 9)
bull 8x ndash 4y = 36bull 3x + 4y = -14bull 11x = 22bull x = 2
bull 2(2) ndash y = 9bull -y = 5bull y = -5
bull (2 -5)
2x ndash y = 93x + 4y = -14
23 = 2 divide 3
Numbers come with Signs (+ -)The sign is in front
Remember These are the same thing
5 ndash 3 = 25 + - 3 = 2
Because Subtraction is Adding the Opposite
If you are confused Circle the number amp the sign in front then do the math
2 ndash 56 + 7 ndash 8 ndash 10 = - 65
ABSOLUTE VALUE
The absolute value is always positive
The absolute value of 5 is 5The absolute value of -5 is 5
To solve drop the bars and make the inside number positive
Watch Out -6 = - 6 because the negative is lurking outside the bars
ALGEBRA OPPOSITES
Opposite of Multiplication is Division
25 4 = 100 100 divide 4 = 25
Positive Negative Numbers
The opposite of -5 is 5
The opposite of 6 is -6
Opposites add to zero
-4 + 4 = 0
Opposite of Addition is
Subtraction12 + 18 = 30
30 ndash 18 = 12
Opposite of Squaring is Square Rooting = 25 = 5
Adding Negative numbers - Think MONEY $$$$$$ You wonrsquot miss it
Algebra Truths
UGLY numbers work the
same as PRETTY Numbers
You canrsquot add FROGS amp
SMILEY FACES
LETTERS amp NUMBERS
work the same
Simplifying is cleaning up
your room put all the
FROGS together amp all the
SMILEY FACES together
then do the Math
No Equal Sign Simplify It Combine Like TermsSimplifying is like cleaning up your room put all the FROGS (Variables) together and all the SMILEY FACES (Numbers) together then do the Math Donrsquot Forget Numbers come with Signs
4x -5 +6x +21 -8x
4x-5
+6x+21
-8x
--------------------------------------------------------------------------------------------------------------------------------4x +6x -8x -5 +21
4x +6x -8x-5 +21
-------------------------------------------------------------------------------------------------------------------------------- 2x +16
2x +16
You canrsquot add FROGS amp SMILEY FACES so you are finished Good Job
WATCH YOUR SIGNS
Adding or Subtracting
If signs are the same add them and use the sign45 + 34 = 79 - 45 ndash 10 = -55
If signs are different subtract and use sign of larger number -18 + 8 = -10 60 ndash 20 = 40
Positive - dirt in the hole
Negative number -digging the hole
ndashPositive number dollars in your pocket Negative number dollars borrowed
Think temperature
Po
siti
ve i
s w
arm
ing
up
Neg
ative is coo
ling
off
Think moneyThink Holes
WATCH YOUR SIGNS
Multiplying or Dividing
If signs are the same answer is positive4 8 = 32 -63 -7 = -9
If signs are different answer is negative-6 7 = -42 -100 10 = -10
Distribution PrincipleMultiply everything in parenthesis by number next
to parenthesis FIRST THING
THINK BIG MOUTH SHOUTING ldquoDO MEDO ME FIRSTrdquo
EXAMPLE 3( 6x ndash 3)
3 (6x ndash 3) = 18x - 9
Distribution PrincipleMultiply everything in parenthesis by number next
to parenthesis FIRST THING
Get them off the busSo they can play football
EXAMPLE 3( 6x ndash 3)
3
18x - 9
6xndash 3
Example 2(10x ndash 3) = 6x + 2
Get the Teams off the Bus2(10x ndash 3) = 6x + 2
Line up the TeamsPenalty for off-sides ndash must change signs
Huddle up 14x = 14
Man on Man Defense 14x = 14 14 14
X = 1
Line of Scrimmage
20x ndash 6 = 6x + 8
20x ndash 6x = 8 + 6
Equation Solving
Think Football ndash Letters Vs Numbers
le rsaquo or lsaquo or geIf the equation has an Inequality sign follow the steps for solving equations with = signs
Play FootballLetters Vrs Numbers
Last Play of the Game
If you have to MULTIPLY or DIVIDE by a negative number Be sure to FLIP the Inequality
NOTICEUGLY NUMBERS WORK THE SAME AS PRETTY NUMBERS
Irsquom not afraid of
Fractions Have
Calculator Will
Calculate
UGLY Numbers Work the Same as PRETTY Numbers
If you can solve 2X + 10 = 40
Then you can solve 5x + 193 = 456
And you can solve
5x + ⅝ = ⅓
Play FootballLetters Vrs Numbers
Clue Words for writing equations from word problems
+
Word Clue
Plusadded to
the sum of increasing by
more than
Word Sentence
1 plus 56 is added to a numberThe sum of 5 and a numberA number is increased by 1015 is more than a number
Algebraic
1+56+x
5+x
x+10
x+15
Addition
Clue Words for writing equations from word problems
AlgebraicWord Clue
Minussubtracted fromthe difference ofdecreased byLessless than
Word Sentence
6 minus 57 subtracted from a numberThe difference of a number and 10A number is decreased by 205 less a number6 less than a number
6-5x-7
x-10
x-205-xx-6
Subtraction __
Clue Words for writing equations from word problems
Word Clue Word Sentence Algebraic
TimesProduct
DoubledTwiceOf (fractions and percents)
7 times a numberProduct of 8 and a numberA number doubledTwice a number12 a number55 of a number
7x = 7x8x
2x2x = 2x12x055x
Multiplication
Word Clue Word Sentence Algebraic
divide
Quotient
divided by
The quotient of a number and 710 divided by a number
xdivide7
10dividex
The first number written before the clue word will be the numerator
Clue Words for writing equations from word problems
Division
1 Consistent ndash one or many solutions2 Inconsistent ndash No solution
1 Independent ndash Only one solution2 Dependent ndash Has infinitely many
solutions
Slope Y ndash Int Graph Type of Systems
of Solutions
Same Same Same line ConsistentDependent
Infinitely many
Same Different Parallel Inconsistent 0
Different DifferentSame
Intersects ConsistentIndependent
1
Consistent and Inconsistent Systems
Slope ndash Intercept Form
y = mx + b
Slope- directions
RiseRun
Y Intercept ndash where to start
Itrsquos a line address
If the slope is a whole number put it on a stick m = 2 slope is 21
To Graph
y = 2X + 1 Starts at 1
Riserun = 21
Directions are up 2 over 1
Example 1 Example 2y = -3X+ 0
y = -3X
Starts at 0
riserun = 3-1
Directions are up 3 over -1
Thanks to httpwwwmathsisfuncomequation_of_linehtml
Linear Equations Standard Form ax + by = cSolving for y Just 3 easy stepsGreenisms Math Terms
1Move x term(Change sides Change signs) AddSubtract x term 2Give x side a hug Parenthesis ( )3Divide by number next to the y Coefficient
Example Solve for Y2x ndash 7y = 12Just 3 easy steps1 -7y = 12 ndash 2x (Move x term (Change sides Change signs)2 -7y = (12-2x) (Give it a hug)3 y = (12-2x) -7 (Divide by number next to the y)
Now you are ready to enter it into the calculator and graph it
WATCH YOUR SIGNS
Example Solve for Y2x ndash 7y = 12
Just 3 easy steps1 -7y = 12 ndash 2x X is offside Penalty change signs2 -7y = (12-2x) Huddle up ( )3 y = (12-2x) -7 Man on man defense
WATCH YOUR SIGNS
Linear Equations Standard Form ax + by = cSolving for y Itrsquos a Football Game
Y VS Everybody Else
Follow football rules
Play FootballY vs everybody else
Now you are ready to enter it into the calculator and graph it
l lines
Same Slope
Slopes are Negative Reciprocal
(Flip amp Change Sign)
PARALLEL LINES
y2 ndash y1
x2 ndash x1
or
y = mx + b
slope
Find Equation of the Line y = mx + b
I need slope (m) amp the y-intercept
(b)
MY ANSWER
y = x +
To find m ndash Solve the equation for y and use mor use the y2 ndash y1
x2 ndash x1 formula
To find b - Plug x y and m into the line equation and solve for b
(Just follow the Rules)
Base xsup2Exponent
Like Bases Exponent Example
Multiply Add
Power up Multiply
Divide Subtract s
Add No change 3asup2 + 5asup2 = 8asup2 asup2 + = asup2 +
Subtract No change 10asup2 - 4asup2 = 6asup2
5a
4
15 9 3a g y
6 4 5
3 7 4a x ka x k
5a
5a
5a
Exponents
=
(xsup2asup3g)(xasup2gsup3) = (xsup3 g )
( gsup3 y) sup3 =
asup3k xsup3
Negative ExponentsJust switch places and make exponent positive
5x5
1x
Example Switch Simplify
5 4 3
2 2 4x g ka x k
4 2
2 5 4 3g x
a x k k
4
2 3 7g
a x k
=
Xsup2
X X∙ = Xsup2 X
X times X is X squared
X
QuadraticsMultiplying Binomials ndash Draw the Face
(x + 8) (x ndash 6)
Multiply Watch your Signs
xsup2 +8x -6x -48
Simplify (Combine Like Terms eat the buggers)xsup2 + 2x -48
Check This First
The Math ldquoFrdquo Word ldquoFactoringrdquo
1 Is there a common factor (number or letter)
NO
Proceed to question 2
Yes
Proceed with CGF
4xsup3 2 2 x x x
4x ( 2xy + xsup2 -3 )
Example
8xsup2y + 4xsup3 - 12x
Factor each term8xsup2y 2 2 2 x x y 12x 2 2 3 x
Circle common terms
8xsup2y 2 2 2 x x y 4xsup3 2 2 x x x 12x 2 2 3 x
Multiply circled numbersThatrsquos your Common Factor
Multiply leftovers put in ( )
2 Perfect squares on end amp 3 terms
xsup2 - 10x + 25
YES SPLIT IT NICE
NO proceed to question 3
xsup2 - 10x + 25Put out baggies to hold the answer (parenthesis)
( x - 5 ) ( x - 5 )
Split the first term nice
Place in baggies
Sign between is same as middle term
Split the second term nice
Place in baggies
ANSWER
(x -5)sup2
3 Perfect squares on end amp 2 terms
YesSame as question 2Except the signsare +-
Example
xsup2 - 64
NO proceed to question 4
( x - 8 ) ( x + 8)
Answer looks like this
4 No perfect squares on end 3 terms amp starts with xsup2
NO proceed to question 5
YES its quadratics in the morning
xsup2 - 10x + 24MA
Multiply to +24 Add to -10
38 11
-3 -8 -11
-2 -12 -14
-6-4 -10 Found it
Put in the parenthesis
(x-6) (x-4)
All Done
2 Search and Seizure (quadratics in the morning)(See question 4)
1 Steal the ldquoardquo and give it to the last term (Multiply)1
5 Is there a number in front of the xsup2 amp does it have 3 terms
Yes Jail BreakNO
Then it is Prime
(canrsquot factor)
Example 2xsup2 + 7x + 3
xsup2 + 7x + 6 (23)
( x + 6 ) ( x + 1 )
3 Arrested and Caught - Divide last terms by ldquoardquo
( x + 62 ) ( x + 12 )
4 Beat it Down - (reduce fractions)
( x + 3 ) ( x + 12 )
5 Parole (kick denominator to the front)(x + 3 ) ( 2 x + 1 )
All Done
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Check This First
Is there a common factor (number or letter)
Greatest Common Factor (GCF)
1 Factor each term2 Circle common terms3 Multiply common terms4 Write it down5 Put out baggie for leftover6 Multiply leftovers for each
term7 Put in baggie ( )
Example
8xsup2y + 4xsup3 - 12x
4x ( 2xy + xsup2 - 3 )
Is there a Square on each End
Perfect Squares
1 Put out Parenthesis2 Split FIRST and LAST numbers nice3 Put in Signs
3 Different Kinds
A xsup2 - 16x + 64
(xndash 8) (x ndash 8)
B xsup2 + 18x + 81
(x + 9) (x + 9)
C xsup2 - 36
(x + 6) (x ndash 6)
Is there a number in front of the xsup2 and does it have 3 terms
Jail Break
1Steal the ldquoardquo and give it to the last term (multiply)2Search and Seizure (Quad in AM)3Arrested and Caught (divide by ldquoardquo)4Beat Down (reduce fractions)5Parole (kick denominator to the front)6Check it out (FACE it)
Example
2xsup2 + 7x + 3
1xsup2 + 7x + 62(x + 6) (x + 1)3(x + 62) (x + 12)4(x + 3) (x +12)5(x + 3) (2x + 1)
No perfect squares and 3 terms and starts with xsup2
Quadratics in the Morning (AM)
1 Make a factor tree2 Multiply to last number add
to middle number3 Put out baggies (parenthesis)4 Split first term nice5 Drop in factors from tree
Example
xsup2 + 12x + 32
(x + 8) (x + 4)
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Solve with Graphing Calculator
1) Solve each equation for y (3 easy steps)
2) Use y= button and enter each equation
3) Use graph to eyeball answerOr
4) Use to find where y1 and y2 are equal
5) Be sure answer is in (x y) form
Y=
Systems of EquationsTo Solve by Graphing
1) Make an x y Chart
2) Select any x solve for y x 2x ndash 4 y 0 2(0) ndash 4 -4 1 2(1) ndash 4 -2
3) Then graph the two points4) Do for both equations5) The answer is where they
cross6) Be sure answer is in (x y)
form
2nd TABLE
bull y + 2x = 9
bull y = 9 ndash 2xbull 3(9 - 2x) ndash 2x = 11
bull 27-6x ndash 2x = 11bull 27 ndash 8x = 11bull - 8x = -16bull x = 2bull y = 9 ndash 2 (2)bull y = 5bull (2 5)
Systems of EquationsSolve by Substitution
(box amp shove)
1) Solve one equation for x or y (change sides change signs)
2) Box it3) Rewrite other equation
and shove box in 4) Solve for surviving letter
1) Distribute2) Combine like terms3) Solve
4) Send it back to box5) Solve for other letter6) Answer in (xy) form
3y -2x = 11y + 2x = 9
Systems of EquationsSolve by Elimination
1) Look for opposite signs
2) Multiply to create opposites
3) Add old and new equation together
4) Solve for surviving letter
5) Plug back into either equation (pick the easy one)
6) Solve for other letter
7) Answer in (xy) format
bull 2x ndash y = 93x + 4y = -14
bull 4(2x ndash y = 9)
bull 8x ndash 4y = 36bull 3x + 4y = -14bull 11x = 22bull x = 2
bull 2(2) ndash y = 9bull -y = 5bull y = -5
bull (2 -5)
2x ndash y = 93x + 4y = -14
Numbers come with Signs (+ -)The sign is in front
Remember These are the same thing
5 ndash 3 = 25 + - 3 = 2
Because Subtraction is Adding the Opposite
If you are confused Circle the number amp the sign in front then do the math
2 ndash 56 + 7 ndash 8 ndash 10 = - 65
ABSOLUTE VALUE
The absolute value is always positive
The absolute value of 5 is 5The absolute value of -5 is 5
To solve drop the bars and make the inside number positive
Watch Out -6 = - 6 because the negative is lurking outside the bars
ALGEBRA OPPOSITES
Opposite of Multiplication is Division
25 4 = 100 100 divide 4 = 25
Positive Negative Numbers
The opposite of -5 is 5
The opposite of 6 is -6
Opposites add to zero
-4 + 4 = 0
Opposite of Addition is
Subtraction12 + 18 = 30
30 ndash 18 = 12
Opposite of Squaring is Square Rooting = 25 = 5
Adding Negative numbers - Think MONEY $$$$$$ You wonrsquot miss it
Algebra Truths
UGLY numbers work the
same as PRETTY Numbers
You canrsquot add FROGS amp
SMILEY FACES
LETTERS amp NUMBERS
work the same
Simplifying is cleaning up
your room put all the
FROGS together amp all the
SMILEY FACES together
then do the Math
No Equal Sign Simplify It Combine Like TermsSimplifying is like cleaning up your room put all the FROGS (Variables) together and all the SMILEY FACES (Numbers) together then do the Math Donrsquot Forget Numbers come with Signs
4x -5 +6x +21 -8x
4x-5
+6x+21
-8x
--------------------------------------------------------------------------------------------------------------------------------4x +6x -8x -5 +21
4x +6x -8x-5 +21
-------------------------------------------------------------------------------------------------------------------------------- 2x +16
2x +16
You canrsquot add FROGS amp SMILEY FACES so you are finished Good Job
WATCH YOUR SIGNS
Adding or Subtracting
If signs are the same add them and use the sign45 + 34 = 79 - 45 ndash 10 = -55
If signs are different subtract and use sign of larger number -18 + 8 = -10 60 ndash 20 = 40
Positive - dirt in the hole
Negative number -digging the hole
ndashPositive number dollars in your pocket Negative number dollars borrowed
Think temperature
Po
siti
ve i
s w
arm
ing
up
Neg
ative is coo
ling
off
Think moneyThink Holes
WATCH YOUR SIGNS
Multiplying or Dividing
If signs are the same answer is positive4 8 = 32 -63 -7 = -9
If signs are different answer is negative-6 7 = -42 -100 10 = -10
Distribution PrincipleMultiply everything in parenthesis by number next
to parenthesis FIRST THING
THINK BIG MOUTH SHOUTING ldquoDO MEDO ME FIRSTrdquo
EXAMPLE 3( 6x ndash 3)
3 (6x ndash 3) = 18x - 9
Distribution PrincipleMultiply everything in parenthesis by number next
to parenthesis FIRST THING
Get them off the busSo they can play football
EXAMPLE 3( 6x ndash 3)
3
18x - 9
6xndash 3
Example 2(10x ndash 3) = 6x + 2
Get the Teams off the Bus2(10x ndash 3) = 6x + 2
Line up the TeamsPenalty for off-sides ndash must change signs
Huddle up 14x = 14
Man on Man Defense 14x = 14 14 14
X = 1
Line of Scrimmage
20x ndash 6 = 6x + 8
20x ndash 6x = 8 + 6
Equation Solving
Think Football ndash Letters Vs Numbers
le rsaquo or lsaquo or geIf the equation has an Inequality sign follow the steps for solving equations with = signs
Play FootballLetters Vrs Numbers
Last Play of the Game
If you have to MULTIPLY or DIVIDE by a negative number Be sure to FLIP the Inequality
NOTICEUGLY NUMBERS WORK THE SAME AS PRETTY NUMBERS
Irsquom not afraid of
Fractions Have
Calculator Will
Calculate
UGLY Numbers Work the Same as PRETTY Numbers
If you can solve 2X + 10 = 40
Then you can solve 5x + 193 = 456
And you can solve
5x + ⅝ = ⅓
Play FootballLetters Vrs Numbers
Clue Words for writing equations from word problems
+
Word Clue
Plusadded to
the sum of increasing by
more than
Word Sentence
1 plus 56 is added to a numberThe sum of 5 and a numberA number is increased by 1015 is more than a number
Algebraic
1+56+x
5+x
x+10
x+15
Addition
Clue Words for writing equations from word problems
AlgebraicWord Clue
Minussubtracted fromthe difference ofdecreased byLessless than
Word Sentence
6 minus 57 subtracted from a numberThe difference of a number and 10A number is decreased by 205 less a number6 less than a number
6-5x-7
x-10
x-205-xx-6
Subtraction __
Clue Words for writing equations from word problems
Word Clue Word Sentence Algebraic
TimesProduct
DoubledTwiceOf (fractions and percents)
7 times a numberProduct of 8 and a numberA number doubledTwice a number12 a number55 of a number
7x = 7x8x
2x2x = 2x12x055x
Multiplication
Word Clue Word Sentence Algebraic
divide
Quotient
divided by
The quotient of a number and 710 divided by a number
xdivide7
10dividex
The first number written before the clue word will be the numerator
Clue Words for writing equations from word problems
Division
1 Consistent ndash one or many solutions2 Inconsistent ndash No solution
1 Independent ndash Only one solution2 Dependent ndash Has infinitely many
solutions
Slope Y ndash Int Graph Type of Systems
of Solutions
Same Same Same line ConsistentDependent
Infinitely many
Same Different Parallel Inconsistent 0
Different DifferentSame
Intersects ConsistentIndependent
1
Consistent and Inconsistent Systems
Slope ndash Intercept Form
y = mx + b
Slope- directions
RiseRun
Y Intercept ndash where to start
Itrsquos a line address
If the slope is a whole number put it on a stick m = 2 slope is 21
To Graph
y = 2X + 1 Starts at 1
Riserun = 21
Directions are up 2 over 1
Example 1 Example 2y = -3X+ 0
y = -3X
Starts at 0
riserun = 3-1
Directions are up 3 over -1
Thanks to httpwwwmathsisfuncomequation_of_linehtml
Linear Equations Standard Form ax + by = cSolving for y Just 3 easy stepsGreenisms Math Terms
1Move x term(Change sides Change signs) AddSubtract x term 2Give x side a hug Parenthesis ( )3Divide by number next to the y Coefficient
Example Solve for Y2x ndash 7y = 12Just 3 easy steps1 -7y = 12 ndash 2x (Move x term (Change sides Change signs)2 -7y = (12-2x) (Give it a hug)3 y = (12-2x) -7 (Divide by number next to the y)
Now you are ready to enter it into the calculator and graph it
WATCH YOUR SIGNS
Example Solve for Y2x ndash 7y = 12
Just 3 easy steps1 -7y = 12 ndash 2x X is offside Penalty change signs2 -7y = (12-2x) Huddle up ( )3 y = (12-2x) -7 Man on man defense
WATCH YOUR SIGNS
Linear Equations Standard Form ax + by = cSolving for y Itrsquos a Football Game
Y VS Everybody Else
Follow football rules
Play FootballY vs everybody else
Now you are ready to enter it into the calculator and graph it
l lines
Same Slope
Slopes are Negative Reciprocal
(Flip amp Change Sign)
PARALLEL LINES
y2 ndash y1
x2 ndash x1
or
y = mx + b
slope
Find Equation of the Line y = mx + b
I need slope (m) amp the y-intercept
(b)
MY ANSWER
y = x +
To find m ndash Solve the equation for y and use mor use the y2 ndash y1
x2 ndash x1 formula
To find b - Plug x y and m into the line equation and solve for b
(Just follow the Rules)
Base xsup2Exponent
Like Bases Exponent Example
Multiply Add
Power up Multiply
Divide Subtract s
Add No change 3asup2 + 5asup2 = 8asup2 asup2 + = asup2 +
Subtract No change 10asup2 - 4asup2 = 6asup2
5a
4
15 9 3a g y
6 4 5
3 7 4a x ka x k
5a
5a
5a
Exponents
=
(xsup2asup3g)(xasup2gsup3) = (xsup3 g )
( gsup3 y) sup3 =
asup3k xsup3
Negative ExponentsJust switch places and make exponent positive
5x5
1x
Example Switch Simplify
5 4 3
2 2 4x g ka x k
4 2
2 5 4 3g x
a x k k
4
2 3 7g
a x k
=
Xsup2
X X∙ = Xsup2 X
X times X is X squared
X
QuadraticsMultiplying Binomials ndash Draw the Face
(x + 8) (x ndash 6)
Multiply Watch your Signs
xsup2 +8x -6x -48
Simplify (Combine Like Terms eat the buggers)xsup2 + 2x -48
Check This First
The Math ldquoFrdquo Word ldquoFactoringrdquo
1 Is there a common factor (number or letter)
NO
Proceed to question 2
Yes
Proceed with CGF
4xsup3 2 2 x x x
4x ( 2xy + xsup2 -3 )
Example
8xsup2y + 4xsup3 - 12x
Factor each term8xsup2y 2 2 2 x x y 12x 2 2 3 x
Circle common terms
8xsup2y 2 2 2 x x y 4xsup3 2 2 x x x 12x 2 2 3 x
Multiply circled numbersThatrsquos your Common Factor
Multiply leftovers put in ( )
2 Perfect squares on end amp 3 terms
xsup2 - 10x + 25
YES SPLIT IT NICE
NO proceed to question 3
xsup2 - 10x + 25Put out baggies to hold the answer (parenthesis)
( x - 5 ) ( x - 5 )
Split the first term nice
Place in baggies
Sign between is same as middle term
Split the second term nice
Place in baggies
ANSWER
(x -5)sup2
3 Perfect squares on end amp 2 terms
YesSame as question 2Except the signsare +-
Example
xsup2 - 64
NO proceed to question 4
( x - 8 ) ( x + 8)
Answer looks like this
4 No perfect squares on end 3 terms amp starts with xsup2
NO proceed to question 5
YES its quadratics in the morning
xsup2 - 10x + 24MA
Multiply to +24 Add to -10
38 11
-3 -8 -11
-2 -12 -14
-6-4 -10 Found it
Put in the parenthesis
(x-6) (x-4)
All Done
2 Search and Seizure (quadratics in the morning)(See question 4)
1 Steal the ldquoardquo and give it to the last term (Multiply)1
5 Is there a number in front of the xsup2 amp does it have 3 terms
Yes Jail BreakNO
Then it is Prime
(canrsquot factor)
Example 2xsup2 + 7x + 3
xsup2 + 7x + 6 (23)
( x + 6 ) ( x + 1 )
3 Arrested and Caught - Divide last terms by ldquoardquo
( x + 62 ) ( x + 12 )
4 Beat it Down - (reduce fractions)
( x + 3 ) ( x + 12 )
5 Parole (kick denominator to the front)(x + 3 ) ( 2 x + 1 )
All Done
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Check This First
Is there a common factor (number or letter)
Greatest Common Factor (GCF)
1 Factor each term2 Circle common terms3 Multiply common terms4 Write it down5 Put out baggie for leftover6 Multiply leftovers for each
term7 Put in baggie ( )
Example
8xsup2y + 4xsup3 - 12x
4x ( 2xy + xsup2 - 3 )
Is there a Square on each End
Perfect Squares
1 Put out Parenthesis2 Split FIRST and LAST numbers nice3 Put in Signs
3 Different Kinds
A xsup2 - 16x + 64
(xndash 8) (x ndash 8)
B xsup2 + 18x + 81
(x + 9) (x + 9)
C xsup2 - 36
(x + 6) (x ndash 6)
Is there a number in front of the xsup2 and does it have 3 terms
Jail Break
1Steal the ldquoardquo and give it to the last term (multiply)2Search and Seizure (Quad in AM)3Arrested and Caught (divide by ldquoardquo)4Beat Down (reduce fractions)5Parole (kick denominator to the front)6Check it out (FACE it)
Example
2xsup2 + 7x + 3
1xsup2 + 7x + 62(x + 6) (x + 1)3(x + 62) (x + 12)4(x + 3) (x +12)5(x + 3) (2x + 1)
No perfect squares and 3 terms and starts with xsup2
Quadratics in the Morning (AM)
1 Make a factor tree2 Multiply to last number add
to middle number3 Put out baggies (parenthesis)4 Split first term nice5 Drop in factors from tree
Example
xsup2 + 12x + 32
(x + 8) (x + 4)
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Solve with Graphing Calculator
1) Solve each equation for y (3 easy steps)
2) Use y= button and enter each equation
3) Use graph to eyeball answerOr
4) Use to find where y1 and y2 are equal
5) Be sure answer is in (x y) form
Y=
Systems of EquationsTo Solve by Graphing
1) Make an x y Chart
2) Select any x solve for y x 2x ndash 4 y 0 2(0) ndash 4 -4 1 2(1) ndash 4 -2
3) Then graph the two points4) Do for both equations5) The answer is where they
cross6) Be sure answer is in (x y)
form
2nd TABLE
bull y + 2x = 9
bull y = 9 ndash 2xbull 3(9 - 2x) ndash 2x = 11
bull 27-6x ndash 2x = 11bull 27 ndash 8x = 11bull - 8x = -16bull x = 2bull y = 9 ndash 2 (2)bull y = 5bull (2 5)
Systems of EquationsSolve by Substitution
(box amp shove)
1) Solve one equation for x or y (change sides change signs)
2) Box it3) Rewrite other equation
and shove box in 4) Solve for surviving letter
1) Distribute2) Combine like terms3) Solve
4) Send it back to box5) Solve for other letter6) Answer in (xy) form
3y -2x = 11y + 2x = 9
Systems of EquationsSolve by Elimination
1) Look for opposite signs
2) Multiply to create opposites
3) Add old and new equation together
4) Solve for surviving letter
5) Plug back into either equation (pick the easy one)
6) Solve for other letter
7) Answer in (xy) format
bull 2x ndash y = 93x + 4y = -14
bull 4(2x ndash y = 9)
bull 8x ndash 4y = 36bull 3x + 4y = -14bull 11x = 22bull x = 2
bull 2(2) ndash y = 9bull -y = 5bull y = -5
bull (2 -5)
2x ndash y = 93x + 4y = -14
ABSOLUTE VALUE
The absolute value is always positive
The absolute value of 5 is 5The absolute value of -5 is 5
To solve drop the bars and make the inside number positive
Watch Out -6 = - 6 because the negative is lurking outside the bars
ALGEBRA OPPOSITES
Opposite of Multiplication is Division
25 4 = 100 100 divide 4 = 25
Positive Negative Numbers
The opposite of -5 is 5
The opposite of 6 is -6
Opposites add to zero
-4 + 4 = 0
Opposite of Addition is
Subtraction12 + 18 = 30
30 ndash 18 = 12
Opposite of Squaring is Square Rooting = 25 = 5
Adding Negative numbers - Think MONEY $$$$$$ You wonrsquot miss it
Algebra Truths
UGLY numbers work the
same as PRETTY Numbers
You canrsquot add FROGS amp
SMILEY FACES
LETTERS amp NUMBERS
work the same
Simplifying is cleaning up
your room put all the
FROGS together amp all the
SMILEY FACES together
then do the Math
No Equal Sign Simplify It Combine Like TermsSimplifying is like cleaning up your room put all the FROGS (Variables) together and all the SMILEY FACES (Numbers) together then do the Math Donrsquot Forget Numbers come with Signs
4x -5 +6x +21 -8x
4x-5
+6x+21
-8x
--------------------------------------------------------------------------------------------------------------------------------4x +6x -8x -5 +21
4x +6x -8x-5 +21
-------------------------------------------------------------------------------------------------------------------------------- 2x +16
2x +16
You canrsquot add FROGS amp SMILEY FACES so you are finished Good Job
WATCH YOUR SIGNS
Adding or Subtracting
If signs are the same add them and use the sign45 + 34 = 79 - 45 ndash 10 = -55
If signs are different subtract and use sign of larger number -18 + 8 = -10 60 ndash 20 = 40
Positive - dirt in the hole
Negative number -digging the hole
ndashPositive number dollars in your pocket Negative number dollars borrowed
Think temperature
Po
siti
ve i
s w
arm
ing
up
Neg
ative is coo
ling
off
Think moneyThink Holes
WATCH YOUR SIGNS
Multiplying or Dividing
If signs are the same answer is positive4 8 = 32 -63 -7 = -9
If signs are different answer is negative-6 7 = -42 -100 10 = -10
Distribution PrincipleMultiply everything in parenthesis by number next
to parenthesis FIRST THING
THINK BIG MOUTH SHOUTING ldquoDO MEDO ME FIRSTrdquo
EXAMPLE 3( 6x ndash 3)
3 (6x ndash 3) = 18x - 9
Distribution PrincipleMultiply everything in parenthesis by number next
to parenthesis FIRST THING
Get them off the busSo they can play football
EXAMPLE 3( 6x ndash 3)
3
18x - 9
6xndash 3
Example 2(10x ndash 3) = 6x + 2
Get the Teams off the Bus2(10x ndash 3) = 6x + 2
Line up the TeamsPenalty for off-sides ndash must change signs
Huddle up 14x = 14
Man on Man Defense 14x = 14 14 14
X = 1
Line of Scrimmage
20x ndash 6 = 6x + 8
20x ndash 6x = 8 + 6
Equation Solving
Think Football ndash Letters Vs Numbers
le rsaquo or lsaquo or geIf the equation has an Inequality sign follow the steps for solving equations with = signs
Play FootballLetters Vrs Numbers
Last Play of the Game
If you have to MULTIPLY or DIVIDE by a negative number Be sure to FLIP the Inequality
NOTICEUGLY NUMBERS WORK THE SAME AS PRETTY NUMBERS
Irsquom not afraid of
Fractions Have
Calculator Will
Calculate
UGLY Numbers Work the Same as PRETTY Numbers
If you can solve 2X + 10 = 40
Then you can solve 5x + 193 = 456
And you can solve
5x + ⅝ = ⅓
Play FootballLetters Vrs Numbers
Clue Words for writing equations from word problems
+
Word Clue
Plusadded to
the sum of increasing by
more than
Word Sentence
1 plus 56 is added to a numberThe sum of 5 and a numberA number is increased by 1015 is more than a number
Algebraic
1+56+x
5+x
x+10
x+15
Addition
Clue Words for writing equations from word problems
AlgebraicWord Clue
Minussubtracted fromthe difference ofdecreased byLessless than
Word Sentence
6 minus 57 subtracted from a numberThe difference of a number and 10A number is decreased by 205 less a number6 less than a number
6-5x-7
x-10
x-205-xx-6
Subtraction __
Clue Words for writing equations from word problems
Word Clue Word Sentence Algebraic
TimesProduct
DoubledTwiceOf (fractions and percents)
7 times a numberProduct of 8 and a numberA number doubledTwice a number12 a number55 of a number
7x = 7x8x
2x2x = 2x12x055x
Multiplication
Word Clue Word Sentence Algebraic
divide
Quotient
divided by
The quotient of a number and 710 divided by a number
xdivide7
10dividex
The first number written before the clue word will be the numerator
Clue Words for writing equations from word problems
Division
1 Consistent ndash one or many solutions2 Inconsistent ndash No solution
1 Independent ndash Only one solution2 Dependent ndash Has infinitely many
solutions
Slope Y ndash Int Graph Type of Systems
of Solutions
Same Same Same line ConsistentDependent
Infinitely many
Same Different Parallel Inconsistent 0
Different DifferentSame
Intersects ConsistentIndependent
1
Consistent and Inconsistent Systems
Slope ndash Intercept Form
y = mx + b
Slope- directions
RiseRun
Y Intercept ndash where to start
Itrsquos a line address
If the slope is a whole number put it on a stick m = 2 slope is 21
To Graph
y = 2X + 1 Starts at 1
Riserun = 21
Directions are up 2 over 1
Example 1 Example 2y = -3X+ 0
y = -3X
Starts at 0
riserun = 3-1
Directions are up 3 over -1
Thanks to httpwwwmathsisfuncomequation_of_linehtml
Linear Equations Standard Form ax + by = cSolving for y Just 3 easy stepsGreenisms Math Terms
1Move x term(Change sides Change signs) AddSubtract x term 2Give x side a hug Parenthesis ( )3Divide by number next to the y Coefficient
Example Solve for Y2x ndash 7y = 12Just 3 easy steps1 -7y = 12 ndash 2x (Move x term (Change sides Change signs)2 -7y = (12-2x) (Give it a hug)3 y = (12-2x) -7 (Divide by number next to the y)
Now you are ready to enter it into the calculator and graph it
WATCH YOUR SIGNS
Example Solve for Y2x ndash 7y = 12
Just 3 easy steps1 -7y = 12 ndash 2x X is offside Penalty change signs2 -7y = (12-2x) Huddle up ( )3 y = (12-2x) -7 Man on man defense
WATCH YOUR SIGNS
Linear Equations Standard Form ax + by = cSolving for y Itrsquos a Football Game
Y VS Everybody Else
Follow football rules
Play FootballY vs everybody else
Now you are ready to enter it into the calculator and graph it
l lines
Same Slope
Slopes are Negative Reciprocal
(Flip amp Change Sign)
PARALLEL LINES
y2 ndash y1
x2 ndash x1
or
y = mx + b
slope
Find Equation of the Line y = mx + b
I need slope (m) amp the y-intercept
(b)
MY ANSWER
y = x +
To find m ndash Solve the equation for y and use mor use the y2 ndash y1
x2 ndash x1 formula
To find b - Plug x y and m into the line equation and solve for b
(Just follow the Rules)
Base xsup2Exponent
Like Bases Exponent Example
Multiply Add
Power up Multiply
Divide Subtract s
Add No change 3asup2 + 5asup2 = 8asup2 asup2 + = asup2 +
Subtract No change 10asup2 - 4asup2 = 6asup2
5a
4
15 9 3a g y
6 4 5
3 7 4a x ka x k
5a
5a
5a
Exponents
=
(xsup2asup3g)(xasup2gsup3) = (xsup3 g )
( gsup3 y) sup3 =
asup3k xsup3
Negative ExponentsJust switch places and make exponent positive
5x5
1x
Example Switch Simplify
5 4 3
2 2 4x g ka x k
4 2
2 5 4 3g x
a x k k
4
2 3 7g
a x k
=
Xsup2
X X∙ = Xsup2 X
X times X is X squared
X
QuadraticsMultiplying Binomials ndash Draw the Face
(x + 8) (x ndash 6)
Multiply Watch your Signs
xsup2 +8x -6x -48
Simplify (Combine Like Terms eat the buggers)xsup2 + 2x -48
Check This First
The Math ldquoFrdquo Word ldquoFactoringrdquo
1 Is there a common factor (number or letter)
NO
Proceed to question 2
Yes
Proceed with CGF
4xsup3 2 2 x x x
4x ( 2xy + xsup2 -3 )
Example
8xsup2y + 4xsup3 - 12x
Factor each term8xsup2y 2 2 2 x x y 12x 2 2 3 x
Circle common terms
8xsup2y 2 2 2 x x y 4xsup3 2 2 x x x 12x 2 2 3 x
Multiply circled numbersThatrsquos your Common Factor
Multiply leftovers put in ( )
2 Perfect squares on end amp 3 terms
xsup2 - 10x + 25
YES SPLIT IT NICE
NO proceed to question 3
xsup2 - 10x + 25Put out baggies to hold the answer (parenthesis)
( x - 5 ) ( x - 5 )
Split the first term nice
Place in baggies
Sign between is same as middle term
Split the second term nice
Place in baggies
ANSWER
(x -5)sup2
3 Perfect squares on end amp 2 terms
YesSame as question 2Except the signsare +-
Example
xsup2 - 64
NO proceed to question 4
( x - 8 ) ( x + 8)
Answer looks like this
4 No perfect squares on end 3 terms amp starts with xsup2
NO proceed to question 5
YES its quadratics in the morning
xsup2 - 10x + 24MA
Multiply to +24 Add to -10
38 11
-3 -8 -11
-2 -12 -14
-6-4 -10 Found it
Put in the parenthesis
(x-6) (x-4)
All Done
2 Search and Seizure (quadratics in the morning)(See question 4)
1 Steal the ldquoardquo and give it to the last term (Multiply)1
5 Is there a number in front of the xsup2 amp does it have 3 terms
Yes Jail BreakNO
Then it is Prime
(canrsquot factor)
Example 2xsup2 + 7x + 3
xsup2 + 7x + 6 (23)
( x + 6 ) ( x + 1 )
3 Arrested and Caught - Divide last terms by ldquoardquo
( x + 62 ) ( x + 12 )
4 Beat it Down - (reduce fractions)
( x + 3 ) ( x + 12 )
5 Parole (kick denominator to the front)(x + 3 ) ( 2 x + 1 )
All Done
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Check This First
Is there a common factor (number or letter)
Greatest Common Factor (GCF)
1 Factor each term2 Circle common terms3 Multiply common terms4 Write it down5 Put out baggie for leftover6 Multiply leftovers for each
term7 Put in baggie ( )
Example
8xsup2y + 4xsup3 - 12x
4x ( 2xy + xsup2 - 3 )
Is there a Square on each End
Perfect Squares
1 Put out Parenthesis2 Split FIRST and LAST numbers nice3 Put in Signs
3 Different Kinds
A xsup2 - 16x + 64
(xndash 8) (x ndash 8)
B xsup2 + 18x + 81
(x + 9) (x + 9)
C xsup2 - 36
(x + 6) (x ndash 6)
Is there a number in front of the xsup2 and does it have 3 terms
Jail Break
1Steal the ldquoardquo and give it to the last term (multiply)2Search and Seizure (Quad in AM)3Arrested and Caught (divide by ldquoardquo)4Beat Down (reduce fractions)5Parole (kick denominator to the front)6Check it out (FACE it)
Example
2xsup2 + 7x + 3
1xsup2 + 7x + 62(x + 6) (x + 1)3(x + 62) (x + 12)4(x + 3) (x +12)5(x + 3) (2x + 1)
No perfect squares and 3 terms and starts with xsup2
Quadratics in the Morning (AM)
1 Make a factor tree2 Multiply to last number add
to middle number3 Put out baggies (parenthesis)4 Split first term nice5 Drop in factors from tree
Example
xsup2 + 12x + 32
(x + 8) (x + 4)
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Solve with Graphing Calculator
1) Solve each equation for y (3 easy steps)
2) Use y= button and enter each equation
3) Use graph to eyeball answerOr
4) Use to find where y1 and y2 are equal
5) Be sure answer is in (x y) form
Y=
Systems of EquationsTo Solve by Graphing
1) Make an x y Chart
2) Select any x solve for y x 2x ndash 4 y 0 2(0) ndash 4 -4 1 2(1) ndash 4 -2
3) Then graph the two points4) Do for both equations5) The answer is where they
cross6) Be sure answer is in (x y)
form
2nd TABLE
bull y + 2x = 9
bull y = 9 ndash 2xbull 3(9 - 2x) ndash 2x = 11
bull 27-6x ndash 2x = 11bull 27 ndash 8x = 11bull - 8x = -16bull x = 2bull y = 9 ndash 2 (2)bull y = 5bull (2 5)
Systems of EquationsSolve by Substitution
(box amp shove)
1) Solve one equation for x or y (change sides change signs)
2) Box it3) Rewrite other equation
and shove box in 4) Solve for surviving letter
1) Distribute2) Combine like terms3) Solve
4) Send it back to box5) Solve for other letter6) Answer in (xy) form
3y -2x = 11y + 2x = 9
Systems of EquationsSolve by Elimination
1) Look for opposite signs
2) Multiply to create opposites
3) Add old and new equation together
4) Solve for surviving letter
5) Plug back into either equation (pick the easy one)
6) Solve for other letter
7) Answer in (xy) format
bull 2x ndash y = 93x + 4y = -14
bull 4(2x ndash y = 9)
bull 8x ndash 4y = 36bull 3x + 4y = -14bull 11x = 22bull x = 2
bull 2(2) ndash y = 9bull -y = 5bull y = -5
bull (2 -5)
2x ndash y = 93x + 4y = -14
ALGEBRA OPPOSITES
Opposite of Multiplication is Division
25 4 = 100 100 divide 4 = 25
Positive Negative Numbers
The opposite of -5 is 5
The opposite of 6 is -6
Opposites add to zero
-4 + 4 = 0
Opposite of Addition is
Subtraction12 + 18 = 30
30 ndash 18 = 12
Opposite of Squaring is Square Rooting = 25 = 5
Adding Negative numbers - Think MONEY $$$$$$ You wonrsquot miss it
Algebra Truths
UGLY numbers work the
same as PRETTY Numbers
You canrsquot add FROGS amp
SMILEY FACES
LETTERS amp NUMBERS
work the same
Simplifying is cleaning up
your room put all the
FROGS together amp all the
SMILEY FACES together
then do the Math
No Equal Sign Simplify It Combine Like TermsSimplifying is like cleaning up your room put all the FROGS (Variables) together and all the SMILEY FACES (Numbers) together then do the Math Donrsquot Forget Numbers come with Signs
4x -5 +6x +21 -8x
4x-5
+6x+21
-8x
--------------------------------------------------------------------------------------------------------------------------------4x +6x -8x -5 +21
4x +6x -8x-5 +21
-------------------------------------------------------------------------------------------------------------------------------- 2x +16
2x +16
You canrsquot add FROGS amp SMILEY FACES so you are finished Good Job
WATCH YOUR SIGNS
Adding or Subtracting
If signs are the same add them and use the sign45 + 34 = 79 - 45 ndash 10 = -55
If signs are different subtract and use sign of larger number -18 + 8 = -10 60 ndash 20 = 40
Positive - dirt in the hole
Negative number -digging the hole
ndashPositive number dollars in your pocket Negative number dollars borrowed
Think temperature
Po
siti
ve i
s w
arm
ing
up
Neg
ative is coo
ling
off
Think moneyThink Holes
WATCH YOUR SIGNS
Multiplying or Dividing
If signs are the same answer is positive4 8 = 32 -63 -7 = -9
If signs are different answer is negative-6 7 = -42 -100 10 = -10
Distribution PrincipleMultiply everything in parenthesis by number next
to parenthesis FIRST THING
THINK BIG MOUTH SHOUTING ldquoDO MEDO ME FIRSTrdquo
EXAMPLE 3( 6x ndash 3)
3 (6x ndash 3) = 18x - 9
Distribution PrincipleMultiply everything in parenthesis by number next
to parenthesis FIRST THING
Get them off the busSo they can play football
EXAMPLE 3( 6x ndash 3)
3
18x - 9
6xndash 3
Example 2(10x ndash 3) = 6x + 2
Get the Teams off the Bus2(10x ndash 3) = 6x + 2
Line up the TeamsPenalty for off-sides ndash must change signs
Huddle up 14x = 14
Man on Man Defense 14x = 14 14 14
X = 1
Line of Scrimmage
20x ndash 6 = 6x + 8
20x ndash 6x = 8 + 6
Equation Solving
Think Football ndash Letters Vs Numbers
le rsaquo or lsaquo or geIf the equation has an Inequality sign follow the steps for solving equations with = signs
Play FootballLetters Vrs Numbers
Last Play of the Game
If you have to MULTIPLY or DIVIDE by a negative number Be sure to FLIP the Inequality
NOTICEUGLY NUMBERS WORK THE SAME AS PRETTY NUMBERS
Irsquom not afraid of
Fractions Have
Calculator Will
Calculate
UGLY Numbers Work the Same as PRETTY Numbers
If you can solve 2X + 10 = 40
Then you can solve 5x + 193 = 456
And you can solve
5x + ⅝ = ⅓
Play FootballLetters Vrs Numbers
Clue Words for writing equations from word problems
+
Word Clue
Plusadded to
the sum of increasing by
more than
Word Sentence
1 plus 56 is added to a numberThe sum of 5 and a numberA number is increased by 1015 is more than a number
Algebraic
1+56+x
5+x
x+10
x+15
Addition
Clue Words for writing equations from word problems
AlgebraicWord Clue
Minussubtracted fromthe difference ofdecreased byLessless than
Word Sentence
6 minus 57 subtracted from a numberThe difference of a number and 10A number is decreased by 205 less a number6 less than a number
6-5x-7
x-10
x-205-xx-6
Subtraction __
Clue Words for writing equations from word problems
Word Clue Word Sentence Algebraic
TimesProduct
DoubledTwiceOf (fractions and percents)
7 times a numberProduct of 8 and a numberA number doubledTwice a number12 a number55 of a number
7x = 7x8x
2x2x = 2x12x055x
Multiplication
Word Clue Word Sentence Algebraic
divide
Quotient
divided by
The quotient of a number and 710 divided by a number
xdivide7
10dividex
The first number written before the clue word will be the numerator
Clue Words for writing equations from word problems
Division
1 Consistent ndash one or many solutions2 Inconsistent ndash No solution
1 Independent ndash Only one solution2 Dependent ndash Has infinitely many
solutions
Slope Y ndash Int Graph Type of Systems
of Solutions
Same Same Same line ConsistentDependent
Infinitely many
Same Different Parallel Inconsistent 0
Different DifferentSame
Intersects ConsistentIndependent
1
Consistent and Inconsistent Systems
Slope ndash Intercept Form
y = mx + b
Slope- directions
RiseRun
Y Intercept ndash where to start
Itrsquos a line address
If the slope is a whole number put it on a stick m = 2 slope is 21
To Graph
y = 2X + 1 Starts at 1
Riserun = 21
Directions are up 2 over 1
Example 1 Example 2y = -3X+ 0
y = -3X
Starts at 0
riserun = 3-1
Directions are up 3 over -1
Thanks to httpwwwmathsisfuncomequation_of_linehtml
Linear Equations Standard Form ax + by = cSolving for y Just 3 easy stepsGreenisms Math Terms
1Move x term(Change sides Change signs) AddSubtract x term 2Give x side a hug Parenthesis ( )3Divide by number next to the y Coefficient
Example Solve for Y2x ndash 7y = 12Just 3 easy steps1 -7y = 12 ndash 2x (Move x term (Change sides Change signs)2 -7y = (12-2x) (Give it a hug)3 y = (12-2x) -7 (Divide by number next to the y)
Now you are ready to enter it into the calculator and graph it
WATCH YOUR SIGNS
Example Solve for Y2x ndash 7y = 12
Just 3 easy steps1 -7y = 12 ndash 2x X is offside Penalty change signs2 -7y = (12-2x) Huddle up ( )3 y = (12-2x) -7 Man on man defense
WATCH YOUR SIGNS
Linear Equations Standard Form ax + by = cSolving for y Itrsquos a Football Game
Y VS Everybody Else
Follow football rules
Play FootballY vs everybody else
Now you are ready to enter it into the calculator and graph it
l lines
Same Slope
Slopes are Negative Reciprocal
(Flip amp Change Sign)
PARALLEL LINES
y2 ndash y1
x2 ndash x1
or
y = mx + b
slope
Find Equation of the Line y = mx + b
I need slope (m) amp the y-intercept
(b)
MY ANSWER
y = x +
To find m ndash Solve the equation for y and use mor use the y2 ndash y1
x2 ndash x1 formula
To find b - Plug x y and m into the line equation and solve for b
(Just follow the Rules)
Base xsup2Exponent
Like Bases Exponent Example
Multiply Add
Power up Multiply
Divide Subtract s
Add No change 3asup2 + 5asup2 = 8asup2 asup2 + = asup2 +
Subtract No change 10asup2 - 4asup2 = 6asup2
5a
4
15 9 3a g y
6 4 5
3 7 4a x ka x k
5a
5a
5a
Exponents
=
(xsup2asup3g)(xasup2gsup3) = (xsup3 g )
( gsup3 y) sup3 =
asup3k xsup3
Negative ExponentsJust switch places and make exponent positive
5x5
1x
Example Switch Simplify
5 4 3
2 2 4x g ka x k
4 2
2 5 4 3g x
a x k k
4
2 3 7g
a x k
=
Xsup2
X X∙ = Xsup2 X
X times X is X squared
X
QuadraticsMultiplying Binomials ndash Draw the Face
(x + 8) (x ndash 6)
Multiply Watch your Signs
xsup2 +8x -6x -48
Simplify (Combine Like Terms eat the buggers)xsup2 + 2x -48
Check This First
The Math ldquoFrdquo Word ldquoFactoringrdquo
1 Is there a common factor (number or letter)
NO
Proceed to question 2
Yes
Proceed with CGF
4xsup3 2 2 x x x
4x ( 2xy + xsup2 -3 )
Example
8xsup2y + 4xsup3 - 12x
Factor each term8xsup2y 2 2 2 x x y 12x 2 2 3 x
Circle common terms
8xsup2y 2 2 2 x x y 4xsup3 2 2 x x x 12x 2 2 3 x
Multiply circled numbersThatrsquos your Common Factor
Multiply leftovers put in ( )
2 Perfect squares on end amp 3 terms
xsup2 - 10x + 25
YES SPLIT IT NICE
NO proceed to question 3
xsup2 - 10x + 25Put out baggies to hold the answer (parenthesis)
( x - 5 ) ( x - 5 )
Split the first term nice
Place in baggies
Sign between is same as middle term
Split the second term nice
Place in baggies
ANSWER
(x -5)sup2
3 Perfect squares on end amp 2 terms
YesSame as question 2Except the signsare +-
Example
xsup2 - 64
NO proceed to question 4
( x - 8 ) ( x + 8)
Answer looks like this
4 No perfect squares on end 3 terms amp starts with xsup2
NO proceed to question 5
YES its quadratics in the morning
xsup2 - 10x + 24MA
Multiply to +24 Add to -10
38 11
-3 -8 -11
-2 -12 -14
-6-4 -10 Found it
Put in the parenthesis
(x-6) (x-4)
All Done
2 Search and Seizure (quadratics in the morning)(See question 4)
1 Steal the ldquoardquo and give it to the last term (Multiply)1
5 Is there a number in front of the xsup2 amp does it have 3 terms
Yes Jail BreakNO
Then it is Prime
(canrsquot factor)
Example 2xsup2 + 7x + 3
xsup2 + 7x + 6 (23)
( x + 6 ) ( x + 1 )
3 Arrested and Caught - Divide last terms by ldquoardquo
( x + 62 ) ( x + 12 )
4 Beat it Down - (reduce fractions)
( x + 3 ) ( x + 12 )
5 Parole (kick denominator to the front)(x + 3 ) ( 2 x + 1 )
All Done
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Check This First
Is there a common factor (number or letter)
Greatest Common Factor (GCF)
1 Factor each term2 Circle common terms3 Multiply common terms4 Write it down5 Put out baggie for leftover6 Multiply leftovers for each
term7 Put in baggie ( )
Example
8xsup2y + 4xsup3 - 12x
4x ( 2xy + xsup2 - 3 )
Is there a Square on each End
Perfect Squares
1 Put out Parenthesis2 Split FIRST and LAST numbers nice3 Put in Signs
3 Different Kinds
A xsup2 - 16x + 64
(xndash 8) (x ndash 8)
B xsup2 + 18x + 81
(x + 9) (x + 9)
C xsup2 - 36
(x + 6) (x ndash 6)
Is there a number in front of the xsup2 and does it have 3 terms
Jail Break
1Steal the ldquoardquo and give it to the last term (multiply)2Search and Seizure (Quad in AM)3Arrested and Caught (divide by ldquoardquo)4Beat Down (reduce fractions)5Parole (kick denominator to the front)6Check it out (FACE it)
Example
2xsup2 + 7x + 3
1xsup2 + 7x + 62(x + 6) (x + 1)3(x + 62) (x + 12)4(x + 3) (x +12)5(x + 3) (2x + 1)
No perfect squares and 3 terms and starts with xsup2
Quadratics in the Morning (AM)
1 Make a factor tree2 Multiply to last number add
to middle number3 Put out baggies (parenthesis)4 Split first term nice5 Drop in factors from tree
Example
xsup2 + 12x + 32
(x + 8) (x + 4)
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Solve with Graphing Calculator
1) Solve each equation for y (3 easy steps)
2) Use y= button and enter each equation
3) Use graph to eyeball answerOr
4) Use to find where y1 and y2 are equal
5) Be sure answer is in (x y) form
Y=
Systems of EquationsTo Solve by Graphing
1) Make an x y Chart
2) Select any x solve for y x 2x ndash 4 y 0 2(0) ndash 4 -4 1 2(1) ndash 4 -2
3) Then graph the two points4) Do for both equations5) The answer is where they
cross6) Be sure answer is in (x y)
form
2nd TABLE
bull y + 2x = 9
bull y = 9 ndash 2xbull 3(9 - 2x) ndash 2x = 11
bull 27-6x ndash 2x = 11bull 27 ndash 8x = 11bull - 8x = -16bull x = 2bull y = 9 ndash 2 (2)bull y = 5bull (2 5)
Systems of EquationsSolve by Substitution
(box amp shove)
1) Solve one equation for x or y (change sides change signs)
2) Box it3) Rewrite other equation
and shove box in 4) Solve for surviving letter
1) Distribute2) Combine like terms3) Solve
4) Send it back to box5) Solve for other letter6) Answer in (xy) form
3y -2x = 11y + 2x = 9
Systems of EquationsSolve by Elimination
1) Look for opposite signs
2) Multiply to create opposites
3) Add old and new equation together
4) Solve for surviving letter
5) Plug back into either equation (pick the easy one)
6) Solve for other letter
7) Answer in (xy) format
bull 2x ndash y = 93x + 4y = -14
bull 4(2x ndash y = 9)
bull 8x ndash 4y = 36bull 3x + 4y = -14bull 11x = 22bull x = 2
bull 2(2) ndash y = 9bull -y = 5bull y = -5
bull (2 -5)
2x ndash y = 93x + 4y = -14
Adding Negative numbers - Think MONEY $$$$$$ You wonrsquot miss it
Algebra Truths
UGLY numbers work the
same as PRETTY Numbers
You canrsquot add FROGS amp
SMILEY FACES
LETTERS amp NUMBERS
work the same
Simplifying is cleaning up
your room put all the
FROGS together amp all the
SMILEY FACES together
then do the Math
No Equal Sign Simplify It Combine Like TermsSimplifying is like cleaning up your room put all the FROGS (Variables) together and all the SMILEY FACES (Numbers) together then do the Math Donrsquot Forget Numbers come with Signs
4x -5 +6x +21 -8x
4x-5
+6x+21
-8x
--------------------------------------------------------------------------------------------------------------------------------4x +6x -8x -5 +21
4x +6x -8x-5 +21
-------------------------------------------------------------------------------------------------------------------------------- 2x +16
2x +16
You canrsquot add FROGS amp SMILEY FACES so you are finished Good Job
WATCH YOUR SIGNS
Adding or Subtracting
If signs are the same add them and use the sign45 + 34 = 79 - 45 ndash 10 = -55
If signs are different subtract and use sign of larger number -18 + 8 = -10 60 ndash 20 = 40
Positive - dirt in the hole
Negative number -digging the hole
ndashPositive number dollars in your pocket Negative number dollars borrowed
Think temperature
Po
siti
ve i
s w
arm
ing
up
Neg
ative is coo
ling
off
Think moneyThink Holes
WATCH YOUR SIGNS
Multiplying or Dividing
If signs are the same answer is positive4 8 = 32 -63 -7 = -9
If signs are different answer is negative-6 7 = -42 -100 10 = -10
Distribution PrincipleMultiply everything in parenthesis by number next
to parenthesis FIRST THING
THINK BIG MOUTH SHOUTING ldquoDO MEDO ME FIRSTrdquo
EXAMPLE 3( 6x ndash 3)
3 (6x ndash 3) = 18x - 9
Distribution PrincipleMultiply everything in parenthesis by number next
to parenthesis FIRST THING
Get them off the busSo they can play football
EXAMPLE 3( 6x ndash 3)
3
18x - 9
6xndash 3
Example 2(10x ndash 3) = 6x + 2
Get the Teams off the Bus2(10x ndash 3) = 6x + 2
Line up the TeamsPenalty for off-sides ndash must change signs
Huddle up 14x = 14
Man on Man Defense 14x = 14 14 14
X = 1
Line of Scrimmage
20x ndash 6 = 6x + 8
20x ndash 6x = 8 + 6
Equation Solving
Think Football ndash Letters Vs Numbers
le rsaquo or lsaquo or geIf the equation has an Inequality sign follow the steps for solving equations with = signs
Play FootballLetters Vrs Numbers
Last Play of the Game
If you have to MULTIPLY or DIVIDE by a negative number Be sure to FLIP the Inequality
NOTICEUGLY NUMBERS WORK THE SAME AS PRETTY NUMBERS
Irsquom not afraid of
Fractions Have
Calculator Will
Calculate
UGLY Numbers Work the Same as PRETTY Numbers
If you can solve 2X + 10 = 40
Then you can solve 5x + 193 = 456
And you can solve
5x + ⅝ = ⅓
Play FootballLetters Vrs Numbers
Clue Words for writing equations from word problems
+
Word Clue
Plusadded to
the sum of increasing by
more than
Word Sentence
1 plus 56 is added to a numberThe sum of 5 and a numberA number is increased by 1015 is more than a number
Algebraic
1+56+x
5+x
x+10
x+15
Addition
Clue Words for writing equations from word problems
AlgebraicWord Clue
Minussubtracted fromthe difference ofdecreased byLessless than
Word Sentence
6 minus 57 subtracted from a numberThe difference of a number and 10A number is decreased by 205 less a number6 less than a number
6-5x-7
x-10
x-205-xx-6
Subtraction __
Clue Words for writing equations from word problems
Word Clue Word Sentence Algebraic
TimesProduct
DoubledTwiceOf (fractions and percents)
7 times a numberProduct of 8 and a numberA number doubledTwice a number12 a number55 of a number
7x = 7x8x
2x2x = 2x12x055x
Multiplication
Word Clue Word Sentence Algebraic
divide
Quotient
divided by
The quotient of a number and 710 divided by a number
xdivide7
10dividex
The first number written before the clue word will be the numerator
Clue Words for writing equations from word problems
Division
1 Consistent ndash one or many solutions2 Inconsistent ndash No solution
1 Independent ndash Only one solution2 Dependent ndash Has infinitely many
solutions
Slope Y ndash Int Graph Type of Systems
of Solutions
Same Same Same line ConsistentDependent
Infinitely many
Same Different Parallel Inconsistent 0
Different DifferentSame
Intersects ConsistentIndependent
1
Consistent and Inconsistent Systems
Slope ndash Intercept Form
y = mx + b
Slope- directions
RiseRun
Y Intercept ndash where to start
Itrsquos a line address
If the slope is a whole number put it on a stick m = 2 slope is 21
To Graph
y = 2X + 1 Starts at 1
Riserun = 21
Directions are up 2 over 1
Example 1 Example 2y = -3X+ 0
y = -3X
Starts at 0
riserun = 3-1
Directions are up 3 over -1
Thanks to httpwwwmathsisfuncomequation_of_linehtml
Linear Equations Standard Form ax + by = cSolving for y Just 3 easy stepsGreenisms Math Terms
1Move x term(Change sides Change signs) AddSubtract x term 2Give x side a hug Parenthesis ( )3Divide by number next to the y Coefficient
Example Solve for Y2x ndash 7y = 12Just 3 easy steps1 -7y = 12 ndash 2x (Move x term (Change sides Change signs)2 -7y = (12-2x) (Give it a hug)3 y = (12-2x) -7 (Divide by number next to the y)
Now you are ready to enter it into the calculator and graph it
WATCH YOUR SIGNS
Example Solve for Y2x ndash 7y = 12
Just 3 easy steps1 -7y = 12 ndash 2x X is offside Penalty change signs2 -7y = (12-2x) Huddle up ( )3 y = (12-2x) -7 Man on man defense
WATCH YOUR SIGNS
Linear Equations Standard Form ax + by = cSolving for y Itrsquos a Football Game
Y VS Everybody Else
Follow football rules
Play FootballY vs everybody else
Now you are ready to enter it into the calculator and graph it
l lines
Same Slope
Slopes are Negative Reciprocal
(Flip amp Change Sign)
PARALLEL LINES
y2 ndash y1
x2 ndash x1
or
y = mx + b
slope
Find Equation of the Line y = mx + b
I need slope (m) amp the y-intercept
(b)
MY ANSWER
y = x +
To find m ndash Solve the equation for y and use mor use the y2 ndash y1
x2 ndash x1 formula
To find b - Plug x y and m into the line equation and solve for b
(Just follow the Rules)
Base xsup2Exponent
Like Bases Exponent Example
Multiply Add
Power up Multiply
Divide Subtract s
Add No change 3asup2 + 5asup2 = 8asup2 asup2 + = asup2 +
Subtract No change 10asup2 - 4asup2 = 6asup2
5a
4
15 9 3a g y
6 4 5
3 7 4a x ka x k
5a
5a
5a
Exponents
=
(xsup2asup3g)(xasup2gsup3) = (xsup3 g )
( gsup3 y) sup3 =
asup3k xsup3
Negative ExponentsJust switch places and make exponent positive
5x5
1x
Example Switch Simplify
5 4 3
2 2 4x g ka x k
4 2
2 5 4 3g x
a x k k
4
2 3 7g
a x k
=
Xsup2
X X∙ = Xsup2 X
X times X is X squared
X
QuadraticsMultiplying Binomials ndash Draw the Face
(x + 8) (x ndash 6)
Multiply Watch your Signs
xsup2 +8x -6x -48
Simplify (Combine Like Terms eat the buggers)xsup2 + 2x -48
Check This First
The Math ldquoFrdquo Word ldquoFactoringrdquo
1 Is there a common factor (number or letter)
NO
Proceed to question 2
Yes
Proceed with CGF
4xsup3 2 2 x x x
4x ( 2xy + xsup2 -3 )
Example
8xsup2y + 4xsup3 - 12x
Factor each term8xsup2y 2 2 2 x x y 12x 2 2 3 x
Circle common terms
8xsup2y 2 2 2 x x y 4xsup3 2 2 x x x 12x 2 2 3 x
Multiply circled numbersThatrsquos your Common Factor
Multiply leftovers put in ( )
2 Perfect squares on end amp 3 terms
xsup2 - 10x + 25
YES SPLIT IT NICE
NO proceed to question 3
xsup2 - 10x + 25Put out baggies to hold the answer (parenthesis)
( x - 5 ) ( x - 5 )
Split the first term nice
Place in baggies
Sign between is same as middle term
Split the second term nice
Place in baggies
ANSWER
(x -5)sup2
3 Perfect squares on end amp 2 terms
YesSame as question 2Except the signsare +-
Example
xsup2 - 64
NO proceed to question 4
( x - 8 ) ( x + 8)
Answer looks like this
4 No perfect squares on end 3 terms amp starts with xsup2
NO proceed to question 5
YES its quadratics in the morning
xsup2 - 10x + 24MA
Multiply to +24 Add to -10
38 11
-3 -8 -11
-2 -12 -14
-6-4 -10 Found it
Put in the parenthesis
(x-6) (x-4)
All Done
2 Search and Seizure (quadratics in the morning)(See question 4)
1 Steal the ldquoardquo and give it to the last term (Multiply)1
5 Is there a number in front of the xsup2 amp does it have 3 terms
Yes Jail BreakNO
Then it is Prime
(canrsquot factor)
Example 2xsup2 + 7x + 3
xsup2 + 7x + 6 (23)
( x + 6 ) ( x + 1 )
3 Arrested and Caught - Divide last terms by ldquoardquo
( x + 62 ) ( x + 12 )
4 Beat it Down - (reduce fractions)
( x + 3 ) ( x + 12 )
5 Parole (kick denominator to the front)(x + 3 ) ( 2 x + 1 )
All Done
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Check This First
Is there a common factor (number or letter)
Greatest Common Factor (GCF)
1 Factor each term2 Circle common terms3 Multiply common terms4 Write it down5 Put out baggie for leftover6 Multiply leftovers for each
term7 Put in baggie ( )
Example
8xsup2y + 4xsup3 - 12x
4x ( 2xy + xsup2 - 3 )
Is there a Square on each End
Perfect Squares
1 Put out Parenthesis2 Split FIRST and LAST numbers nice3 Put in Signs
3 Different Kinds
A xsup2 - 16x + 64
(xndash 8) (x ndash 8)
B xsup2 + 18x + 81
(x + 9) (x + 9)
C xsup2 - 36
(x + 6) (x ndash 6)
Is there a number in front of the xsup2 and does it have 3 terms
Jail Break
1Steal the ldquoardquo and give it to the last term (multiply)2Search and Seizure (Quad in AM)3Arrested and Caught (divide by ldquoardquo)4Beat Down (reduce fractions)5Parole (kick denominator to the front)6Check it out (FACE it)
Example
2xsup2 + 7x + 3
1xsup2 + 7x + 62(x + 6) (x + 1)3(x + 62) (x + 12)4(x + 3) (x +12)5(x + 3) (2x + 1)
No perfect squares and 3 terms and starts with xsup2
Quadratics in the Morning (AM)
1 Make a factor tree2 Multiply to last number add
to middle number3 Put out baggies (parenthesis)4 Split first term nice5 Drop in factors from tree
Example
xsup2 + 12x + 32
(x + 8) (x + 4)
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Solve with Graphing Calculator
1) Solve each equation for y (3 easy steps)
2) Use y= button and enter each equation
3) Use graph to eyeball answerOr
4) Use to find where y1 and y2 are equal
5) Be sure answer is in (x y) form
Y=
Systems of EquationsTo Solve by Graphing
1) Make an x y Chart
2) Select any x solve for y x 2x ndash 4 y 0 2(0) ndash 4 -4 1 2(1) ndash 4 -2
3) Then graph the two points4) Do for both equations5) The answer is where they
cross6) Be sure answer is in (x y)
form
2nd TABLE
bull y + 2x = 9
bull y = 9 ndash 2xbull 3(9 - 2x) ndash 2x = 11
bull 27-6x ndash 2x = 11bull 27 ndash 8x = 11bull - 8x = -16bull x = 2bull y = 9 ndash 2 (2)bull y = 5bull (2 5)
Systems of EquationsSolve by Substitution
(box amp shove)
1) Solve one equation for x or y (change sides change signs)
2) Box it3) Rewrite other equation
and shove box in 4) Solve for surviving letter
1) Distribute2) Combine like terms3) Solve
4) Send it back to box5) Solve for other letter6) Answer in (xy) form
3y -2x = 11y + 2x = 9
Systems of EquationsSolve by Elimination
1) Look for opposite signs
2) Multiply to create opposites
3) Add old and new equation together
4) Solve for surviving letter
5) Plug back into either equation (pick the easy one)
6) Solve for other letter
7) Answer in (xy) format
bull 2x ndash y = 93x + 4y = -14
bull 4(2x ndash y = 9)
bull 8x ndash 4y = 36bull 3x + 4y = -14bull 11x = 22bull x = 2
bull 2(2) ndash y = 9bull -y = 5bull y = -5
bull (2 -5)
2x ndash y = 93x + 4y = -14
No Equal Sign Simplify It Combine Like TermsSimplifying is like cleaning up your room put all the FROGS (Variables) together and all the SMILEY FACES (Numbers) together then do the Math Donrsquot Forget Numbers come with Signs
4x -5 +6x +21 -8x
4x-5
+6x+21
-8x
--------------------------------------------------------------------------------------------------------------------------------4x +6x -8x -5 +21
4x +6x -8x-5 +21
-------------------------------------------------------------------------------------------------------------------------------- 2x +16
2x +16
You canrsquot add FROGS amp SMILEY FACES so you are finished Good Job
WATCH YOUR SIGNS
Adding or Subtracting
If signs are the same add them and use the sign45 + 34 = 79 - 45 ndash 10 = -55
If signs are different subtract and use sign of larger number -18 + 8 = -10 60 ndash 20 = 40
Positive - dirt in the hole
Negative number -digging the hole
ndashPositive number dollars in your pocket Negative number dollars borrowed
Think temperature
Po
siti
ve i
s w
arm
ing
up
Neg
ative is coo
ling
off
Think moneyThink Holes
WATCH YOUR SIGNS
Multiplying or Dividing
If signs are the same answer is positive4 8 = 32 -63 -7 = -9
If signs are different answer is negative-6 7 = -42 -100 10 = -10
Distribution PrincipleMultiply everything in parenthesis by number next
to parenthesis FIRST THING
THINK BIG MOUTH SHOUTING ldquoDO MEDO ME FIRSTrdquo
EXAMPLE 3( 6x ndash 3)
3 (6x ndash 3) = 18x - 9
Distribution PrincipleMultiply everything in parenthesis by number next
to parenthesis FIRST THING
Get them off the busSo they can play football
EXAMPLE 3( 6x ndash 3)
3
18x - 9
6xndash 3
Example 2(10x ndash 3) = 6x + 2
Get the Teams off the Bus2(10x ndash 3) = 6x + 2
Line up the TeamsPenalty for off-sides ndash must change signs
Huddle up 14x = 14
Man on Man Defense 14x = 14 14 14
X = 1
Line of Scrimmage
20x ndash 6 = 6x + 8
20x ndash 6x = 8 + 6
Equation Solving
Think Football ndash Letters Vs Numbers
le rsaquo or lsaquo or geIf the equation has an Inequality sign follow the steps for solving equations with = signs
Play FootballLetters Vrs Numbers
Last Play of the Game
If you have to MULTIPLY or DIVIDE by a negative number Be sure to FLIP the Inequality
NOTICEUGLY NUMBERS WORK THE SAME AS PRETTY NUMBERS
Irsquom not afraid of
Fractions Have
Calculator Will
Calculate
UGLY Numbers Work the Same as PRETTY Numbers
If you can solve 2X + 10 = 40
Then you can solve 5x + 193 = 456
And you can solve
5x + ⅝ = ⅓
Play FootballLetters Vrs Numbers
Clue Words for writing equations from word problems
+
Word Clue
Plusadded to
the sum of increasing by
more than
Word Sentence
1 plus 56 is added to a numberThe sum of 5 and a numberA number is increased by 1015 is more than a number
Algebraic
1+56+x
5+x
x+10
x+15
Addition
Clue Words for writing equations from word problems
AlgebraicWord Clue
Minussubtracted fromthe difference ofdecreased byLessless than
Word Sentence
6 minus 57 subtracted from a numberThe difference of a number and 10A number is decreased by 205 less a number6 less than a number
6-5x-7
x-10
x-205-xx-6
Subtraction __
Clue Words for writing equations from word problems
Word Clue Word Sentence Algebraic
TimesProduct
DoubledTwiceOf (fractions and percents)
7 times a numberProduct of 8 and a numberA number doubledTwice a number12 a number55 of a number
7x = 7x8x
2x2x = 2x12x055x
Multiplication
Word Clue Word Sentence Algebraic
divide
Quotient
divided by
The quotient of a number and 710 divided by a number
xdivide7
10dividex
The first number written before the clue word will be the numerator
Clue Words for writing equations from word problems
Division
1 Consistent ndash one or many solutions2 Inconsistent ndash No solution
1 Independent ndash Only one solution2 Dependent ndash Has infinitely many
solutions
Slope Y ndash Int Graph Type of Systems
of Solutions
Same Same Same line ConsistentDependent
Infinitely many
Same Different Parallel Inconsistent 0
Different DifferentSame
Intersects ConsistentIndependent
1
Consistent and Inconsistent Systems
Slope ndash Intercept Form
y = mx + b
Slope- directions
RiseRun
Y Intercept ndash where to start
Itrsquos a line address
If the slope is a whole number put it on a stick m = 2 slope is 21
To Graph
y = 2X + 1 Starts at 1
Riserun = 21
Directions are up 2 over 1
Example 1 Example 2y = -3X+ 0
y = -3X
Starts at 0
riserun = 3-1
Directions are up 3 over -1
Thanks to httpwwwmathsisfuncomequation_of_linehtml
Linear Equations Standard Form ax + by = cSolving for y Just 3 easy stepsGreenisms Math Terms
1Move x term(Change sides Change signs) AddSubtract x term 2Give x side a hug Parenthesis ( )3Divide by number next to the y Coefficient
Example Solve for Y2x ndash 7y = 12Just 3 easy steps1 -7y = 12 ndash 2x (Move x term (Change sides Change signs)2 -7y = (12-2x) (Give it a hug)3 y = (12-2x) -7 (Divide by number next to the y)
Now you are ready to enter it into the calculator and graph it
WATCH YOUR SIGNS
Example Solve for Y2x ndash 7y = 12
Just 3 easy steps1 -7y = 12 ndash 2x X is offside Penalty change signs2 -7y = (12-2x) Huddle up ( )3 y = (12-2x) -7 Man on man defense
WATCH YOUR SIGNS
Linear Equations Standard Form ax + by = cSolving for y Itrsquos a Football Game
Y VS Everybody Else
Follow football rules
Play FootballY vs everybody else
Now you are ready to enter it into the calculator and graph it
l lines
Same Slope
Slopes are Negative Reciprocal
(Flip amp Change Sign)
PARALLEL LINES
y2 ndash y1
x2 ndash x1
or
y = mx + b
slope
Find Equation of the Line y = mx + b
I need slope (m) amp the y-intercept
(b)
MY ANSWER
y = x +
To find m ndash Solve the equation for y and use mor use the y2 ndash y1
x2 ndash x1 formula
To find b - Plug x y and m into the line equation and solve for b
(Just follow the Rules)
Base xsup2Exponent
Like Bases Exponent Example
Multiply Add
Power up Multiply
Divide Subtract s
Add No change 3asup2 + 5asup2 = 8asup2 asup2 + = asup2 +
Subtract No change 10asup2 - 4asup2 = 6asup2
5a
4
15 9 3a g y
6 4 5
3 7 4a x ka x k
5a
5a
5a
Exponents
=
(xsup2asup3g)(xasup2gsup3) = (xsup3 g )
( gsup3 y) sup3 =
asup3k xsup3
Negative ExponentsJust switch places and make exponent positive
5x5
1x
Example Switch Simplify
5 4 3
2 2 4x g ka x k
4 2
2 5 4 3g x
a x k k
4
2 3 7g
a x k
=
Xsup2
X X∙ = Xsup2 X
X times X is X squared
X
QuadraticsMultiplying Binomials ndash Draw the Face
(x + 8) (x ndash 6)
Multiply Watch your Signs
xsup2 +8x -6x -48
Simplify (Combine Like Terms eat the buggers)xsup2 + 2x -48
Check This First
The Math ldquoFrdquo Word ldquoFactoringrdquo
1 Is there a common factor (number or letter)
NO
Proceed to question 2
Yes
Proceed with CGF
4xsup3 2 2 x x x
4x ( 2xy + xsup2 -3 )
Example
8xsup2y + 4xsup3 - 12x
Factor each term8xsup2y 2 2 2 x x y 12x 2 2 3 x
Circle common terms
8xsup2y 2 2 2 x x y 4xsup3 2 2 x x x 12x 2 2 3 x
Multiply circled numbersThatrsquos your Common Factor
Multiply leftovers put in ( )
2 Perfect squares on end amp 3 terms
xsup2 - 10x + 25
YES SPLIT IT NICE
NO proceed to question 3
xsup2 - 10x + 25Put out baggies to hold the answer (parenthesis)
( x - 5 ) ( x - 5 )
Split the first term nice
Place in baggies
Sign between is same as middle term
Split the second term nice
Place in baggies
ANSWER
(x -5)sup2
3 Perfect squares on end amp 2 terms
YesSame as question 2Except the signsare +-
Example
xsup2 - 64
NO proceed to question 4
( x - 8 ) ( x + 8)
Answer looks like this
4 No perfect squares on end 3 terms amp starts with xsup2
NO proceed to question 5
YES its quadratics in the morning
xsup2 - 10x + 24MA
Multiply to +24 Add to -10
38 11
-3 -8 -11
-2 -12 -14
-6-4 -10 Found it
Put in the parenthesis
(x-6) (x-4)
All Done
2 Search and Seizure (quadratics in the morning)(See question 4)
1 Steal the ldquoardquo and give it to the last term (Multiply)1
5 Is there a number in front of the xsup2 amp does it have 3 terms
Yes Jail BreakNO
Then it is Prime
(canrsquot factor)
Example 2xsup2 + 7x + 3
xsup2 + 7x + 6 (23)
( x + 6 ) ( x + 1 )
3 Arrested and Caught - Divide last terms by ldquoardquo
( x + 62 ) ( x + 12 )
4 Beat it Down - (reduce fractions)
( x + 3 ) ( x + 12 )
5 Parole (kick denominator to the front)(x + 3 ) ( 2 x + 1 )
All Done
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Check This First
Is there a common factor (number or letter)
Greatest Common Factor (GCF)
1 Factor each term2 Circle common terms3 Multiply common terms4 Write it down5 Put out baggie for leftover6 Multiply leftovers for each
term7 Put in baggie ( )
Example
8xsup2y + 4xsup3 - 12x
4x ( 2xy + xsup2 - 3 )
Is there a Square on each End
Perfect Squares
1 Put out Parenthesis2 Split FIRST and LAST numbers nice3 Put in Signs
3 Different Kinds
A xsup2 - 16x + 64
(xndash 8) (x ndash 8)
B xsup2 + 18x + 81
(x + 9) (x + 9)
C xsup2 - 36
(x + 6) (x ndash 6)
Is there a number in front of the xsup2 and does it have 3 terms
Jail Break
1Steal the ldquoardquo and give it to the last term (multiply)2Search and Seizure (Quad in AM)3Arrested and Caught (divide by ldquoardquo)4Beat Down (reduce fractions)5Parole (kick denominator to the front)6Check it out (FACE it)
Example
2xsup2 + 7x + 3
1xsup2 + 7x + 62(x + 6) (x + 1)3(x + 62) (x + 12)4(x + 3) (x +12)5(x + 3) (2x + 1)
No perfect squares and 3 terms and starts with xsup2
Quadratics in the Morning (AM)
1 Make a factor tree2 Multiply to last number add
to middle number3 Put out baggies (parenthesis)4 Split first term nice5 Drop in factors from tree
Example
xsup2 + 12x + 32
(x + 8) (x + 4)
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Solve with Graphing Calculator
1) Solve each equation for y (3 easy steps)
2) Use y= button and enter each equation
3) Use graph to eyeball answerOr
4) Use to find where y1 and y2 are equal
5) Be sure answer is in (x y) form
Y=
Systems of EquationsTo Solve by Graphing
1) Make an x y Chart
2) Select any x solve for y x 2x ndash 4 y 0 2(0) ndash 4 -4 1 2(1) ndash 4 -2
3) Then graph the two points4) Do for both equations5) The answer is where they
cross6) Be sure answer is in (x y)
form
2nd TABLE
bull y + 2x = 9
bull y = 9 ndash 2xbull 3(9 - 2x) ndash 2x = 11
bull 27-6x ndash 2x = 11bull 27 ndash 8x = 11bull - 8x = -16bull x = 2bull y = 9 ndash 2 (2)bull y = 5bull (2 5)
Systems of EquationsSolve by Substitution
(box amp shove)
1) Solve one equation for x or y (change sides change signs)
2) Box it3) Rewrite other equation
and shove box in 4) Solve for surviving letter
1) Distribute2) Combine like terms3) Solve
4) Send it back to box5) Solve for other letter6) Answer in (xy) form
3y -2x = 11y + 2x = 9
Systems of EquationsSolve by Elimination
1) Look for opposite signs
2) Multiply to create opposites
3) Add old and new equation together
4) Solve for surviving letter
5) Plug back into either equation (pick the easy one)
6) Solve for other letter
7) Answer in (xy) format
bull 2x ndash y = 93x + 4y = -14
bull 4(2x ndash y = 9)
bull 8x ndash 4y = 36bull 3x + 4y = -14bull 11x = 22bull x = 2
bull 2(2) ndash y = 9bull -y = 5bull y = -5
bull (2 -5)
2x ndash y = 93x + 4y = -14
WATCH YOUR SIGNS
Adding or Subtracting
If signs are the same add them and use the sign45 + 34 = 79 - 45 ndash 10 = -55
If signs are different subtract and use sign of larger number -18 + 8 = -10 60 ndash 20 = 40
Positive - dirt in the hole
Negative number -digging the hole
ndashPositive number dollars in your pocket Negative number dollars borrowed
Think temperature
Po
siti
ve i
s w
arm
ing
up
Neg
ative is coo
ling
off
Think moneyThink Holes
WATCH YOUR SIGNS
Multiplying or Dividing
If signs are the same answer is positive4 8 = 32 -63 -7 = -9
If signs are different answer is negative-6 7 = -42 -100 10 = -10
Distribution PrincipleMultiply everything in parenthesis by number next
to parenthesis FIRST THING
THINK BIG MOUTH SHOUTING ldquoDO MEDO ME FIRSTrdquo
EXAMPLE 3( 6x ndash 3)
3 (6x ndash 3) = 18x - 9
Distribution PrincipleMultiply everything in parenthesis by number next
to parenthesis FIRST THING
Get them off the busSo they can play football
EXAMPLE 3( 6x ndash 3)
3
18x - 9
6xndash 3
Example 2(10x ndash 3) = 6x + 2
Get the Teams off the Bus2(10x ndash 3) = 6x + 2
Line up the TeamsPenalty for off-sides ndash must change signs
Huddle up 14x = 14
Man on Man Defense 14x = 14 14 14
X = 1
Line of Scrimmage
20x ndash 6 = 6x + 8
20x ndash 6x = 8 + 6
Equation Solving
Think Football ndash Letters Vs Numbers
le rsaquo or lsaquo or geIf the equation has an Inequality sign follow the steps for solving equations with = signs
Play FootballLetters Vrs Numbers
Last Play of the Game
If you have to MULTIPLY or DIVIDE by a negative number Be sure to FLIP the Inequality
NOTICEUGLY NUMBERS WORK THE SAME AS PRETTY NUMBERS
Irsquom not afraid of
Fractions Have
Calculator Will
Calculate
UGLY Numbers Work the Same as PRETTY Numbers
If you can solve 2X + 10 = 40
Then you can solve 5x + 193 = 456
And you can solve
5x + ⅝ = ⅓
Play FootballLetters Vrs Numbers
Clue Words for writing equations from word problems
+
Word Clue
Plusadded to
the sum of increasing by
more than
Word Sentence
1 plus 56 is added to a numberThe sum of 5 and a numberA number is increased by 1015 is more than a number
Algebraic
1+56+x
5+x
x+10
x+15
Addition
Clue Words for writing equations from word problems
AlgebraicWord Clue
Minussubtracted fromthe difference ofdecreased byLessless than
Word Sentence
6 minus 57 subtracted from a numberThe difference of a number and 10A number is decreased by 205 less a number6 less than a number
6-5x-7
x-10
x-205-xx-6
Subtraction __
Clue Words for writing equations from word problems
Word Clue Word Sentence Algebraic
TimesProduct
DoubledTwiceOf (fractions and percents)
7 times a numberProduct of 8 and a numberA number doubledTwice a number12 a number55 of a number
7x = 7x8x
2x2x = 2x12x055x
Multiplication
Word Clue Word Sentence Algebraic
divide
Quotient
divided by
The quotient of a number and 710 divided by a number
xdivide7
10dividex
The first number written before the clue word will be the numerator
Clue Words for writing equations from word problems
Division
1 Consistent ndash one or many solutions2 Inconsistent ndash No solution
1 Independent ndash Only one solution2 Dependent ndash Has infinitely many
solutions
Slope Y ndash Int Graph Type of Systems
of Solutions
Same Same Same line ConsistentDependent
Infinitely many
Same Different Parallel Inconsistent 0
Different DifferentSame
Intersects ConsistentIndependent
1
Consistent and Inconsistent Systems
Slope ndash Intercept Form
y = mx + b
Slope- directions
RiseRun
Y Intercept ndash where to start
Itrsquos a line address
If the slope is a whole number put it on a stick m = 2 slope is 21
To Graph
y = 2X + 1 Starts at 1
Riserun = 21
Directions are up 2 over 1
Example 1 Example 2y = -3X+ 0
y = -3X
Starts at 0
riserun = 3-1
Directions are up 3 over -1
Thanks to httpwwwmathsisfuncomequation_of_linehtml
Linear Equations Standard Form ax + by = cSolving for y Just 3 easy stepsGreenisms Math Terms
1Move x term(Change sides Change signs) AddSubtract x term 2Give x side a hug Parenthesis ( )3Divide by number next to the y Coefficient
Example Solve for Y2x ndash 7y = 12Just 3 easy steps1 -7y = 12 ndash 2x (Move x term (Change sides Change signs)2 -7y = (12-2x) (Give it a hug)3 y = (12-2x) -7 (Divide by number next to the y)
Now you are ready to enter it into the calculator and graph it
WATCH YOUR SIGNS
Example Solve for Y2x ndash 7y = 12
Just 3 easy steps1 -7y = 12 ndash 2x X is offside Penalty change signs2 -7y = (12-2x) Huddle up ( )3 y = (12-2x) -7 Man on man defense
WATCH YOUR SIGNS
Linear Equations Standard Form ax + by = cSolving for y Itrsquos a Football Game
Y VS Everybody Else
Follow football rules
Play FootballY vs everybody else
Now you are ready to enter it into the calculator and graph it
l lines
Same Slope
Slopes are Negative Reciprocal
(Flip amp Change Sign)
PARALLEL LINES
y2 ndash y1
x2 ndash x1
or
y = mx + b
slope
Find Equation of the Line y = mx + b
I need slope (m) amp the y-intercept
(b)
MY ANSWER
y = x +
To find m ndash Solve the equation for y and use mor use the y2 ndash y1
x2 ndash x1 formula
To find b - Plug x y and m into the line equation and solve for b
(Just follow the Rules)
Base xsup2Exponent
Like Bases Exponent Example
Multiply Add
Power up Multiply
Divide Subtract s
Add No change 3asup2 + 5asup2 = 8asup2 asup2 + = asup2 +
Subtract No change 10asup2 - 4asup2 = 6asup2
5a
4
15 9 3a g y
6 4 5
3 7 4a x ka x k
5a
5a
5a
Exponents
=
(xsup2asup3g)(xasup2gsup3) = (xsup3 g )
( gsup3 y) sup3 =
asup3k xsup3
Negative ExponentsJust switch places and make exponent positive
5x5
1x
Example Switch Simplify
5 4 3
2 2 4x g ka x k
4 2
2 5 4 3g x
a x k k
4
2 3 7g
a x k
=
Xsup2
X X∙ = Xsup2 X
X times X is X squared
X
QuadraticsMultiplying Binomials ndash Draw the Face
(x + 8) (x ndash 6)
Multiply Watch your Signs
xsup2 +8x -6x -48
Simplify (Combine Like Terms eat the buggers)xsup2 + 2x -48
Check This First
The Math ldquoFrdquo Word ldquoFactoringrdquo
1 Is there a common factor (number or letter)
NO
Proceed to question 2
Yes
Proceed with CGF
4xsup3 2 2 x x x
4x ( 2xy + xsup2 -3 )
Example
8xsup2y + 4xsup3 - 12x
Factor each term8xsup2y 2 2 2 x x y 12x 2 2 3 x
Circle common terms
8xsup2y 2 2 2 x x y 4xsup3 2 2 x x x 12x 2 2 3 x
Multiply circled numbersThatrsquos your Common Factor
Multiply leftovers put in ( )
2 Perfect squares on end amp 3 terms
xsup2 - 10x + 25
YES SPLIT IT NICE
NO proceed to question 3
xsup2 - 10x + 25Put out baggies to hold the answer (parenthesis)
( x - 5 ) ( x - 5 )
Split the first term nice
Place in baggies
Sign between is same as middle term
Split the second term nice
Place in baggies
ANSWER
(x -5)sup2
3 Perfect squares on end amp 2 terms
YesSame as question 2Except the signsare +-
Example
xsup2 - 64
NO proceed to question 4
( x - 8 ) ( x + 8)
Answer looks like this
4 No perfect squares on end 3 terms amp starts with xsup2
NO proceed to question 5
YES its quadratics in the morning
xsup2 - 10x + 24MA
Multiply to +24 Add to -10
38 11
-3 -8 -11
-2 -12 -14
-6-4 -10 Found it
Put in the parenthesis
(x-6) (x-4)
All Done
2 Search and Seizure (quadratics in the morning)(See question 4)
1 Steal the ldquoardquo and give it to the last term (Multiply)1
5 Is there a number in front of the xsup2 amp does it have 3 terms
Yes Jail BreakNO
Then it is Prime
(canrsquot factor)
Example 2xsup2 + 7x + 3
xsup2 + 7x + 6 (23)
( x + 6 ) ( x + 1 )
3 Arrested and Caught - Divide last terms by ldquoardquo
( x + 62 ) ( x + 12 )
4 Beat it Down - (reduce fractions)
( x + 3 ) ( x + 12 )
5 Parole (kick denominator to the front)(x + 3 ) ( 2 x + 1 )
All Done
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Check This First
Is there a common factor (number or letter)
Greatest Common Factor (GCF)
1 Factor each term2 Circle common terms3 Multiply common terms4 Write it down5 Put out baggie for leftover6 Multiply leftovers for each
term7 Put in baggie ( )
Example
8xsup2y + 4xsup3 - 12x
4x ( 2xy + xsup2 - 3 )
Is there a Square on each End
Perfect Squares
1 Put out Parenthesis2 Split FIRST and LAST numbers nice3 Put in Signs
3 Different Kinds
A xsup2 - 16x + 64
(xndash 8) (x ndash 8)
B xsup2 + 18x + 81
(x + 9) (x + 9)
C xsup2 - 36
(x + 6) (x ndash 6)
Is there a number in front of the xsup2 and does it have 3 terms
Jail Break
1Steal the ldquoardquo and give it to the last term (multiply)2Search and Seizure (Quad in AM)3Arrested and Caught (divide by ldquoardquo)4Beat Down (reduce fractions)5Parole (kick denominator to the front)6Check it out (FACE it)
Example
2xsup2 + 7x + 3
1xsup2 + 7x + 62(x + 6) (x + 1)3(x + 62) (x + 12)4(x + 3) (x +12)5(x + 3) (2x + 1)
No perfect squares and 3 terms and starts with xsup2
Quadratics in the Morning (AM)
1 Make a factor tree2 Multiply to last number add
to middle number3 Put out baggies (parenthesis)4 Split first term nice5 Drop in factors from tree
Example
xsup2 + 12x + 32
(x + 8) (x + 4)
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Solve with Graphing Calculator
1) Solve each equation for y (3 easy steps)
2) Use y= button and enter each equation
3) Use graph to eyeball answerOr
4) Use to find where y1 and y2 are equal
5) Be sure answer is in (x y) form
Y=
Systems of EquationsTo Solve by Graphing
1) Make an x y Chart
2) Select any x solve for y x 2x ndash 4 y 0 2(0) ndash 4 -4 1 2(1) ndash 4 -2
3) Then graph the two points4) Do for both equations5) The answer is where they
cross6) Be sure answer is in (x y)
form
2nd TABLE
bull y + 2x = 9
bull y = 9 ndash 2xbull 3(9 - 2x) ndash 2x = 11
bull 27-6x ndash 2x = 11bull 27 ndash 8x = 11bull - 8x = -16bull x = 2bull y = 9 ndash 2 (2)bull y = 5bull (2 5)
Systems of EquationsSolve by Substitution
(box amp shove)
1) Solve one equation for x or y (change sides change signs)
2) Box it3) Rewrite other equation
and shove box in 4) Solve for surviving letter
1) Distribute2) Combine like terms3) Solve
4) Send it back to box5) Solve for other letter6) Answer in (xy) form
3y -2x = 11y + 2x = 9
Systems of EquationsSolve by Elimination
1) Look for opposite signs
2) Multiply to create opposites
3) Add old and new equation together
4) Solve for surviving letter
5) Plug back into either equation (pick the easy one)
6) Solve for other letter
7) Answer in (xy) format
bull 2x ndash y = 93x + 4y = -14
bull 4(2x ndash y = 9)
bull 8x ndash 4y = 36bull 3x + 4y = -14bull 11x = 22bull x = 2
bull 2(2) ndash y = 9bull -y = 5bull y = -5
bull (2 -5)
2x ndash y = 93x + 4y = -14
WATCH YOUR SIGNS
Multiplying or Dividing
If signs are the same answer is positive4 8 = 32 -63 -7 = -9
If signs are different answer is negative-6 7 = -42 -100 10 = -10
Distribution PrincipleMultiply everything in parenthesis by number next
to parenthesis FIRST THING
THINK BIG MOUTH SHOUTING ldquoDO MEDO ME FIRSTrdquo
EXAMPLE 3( 6x ndash 3)
3 (6x ndash 3) = 18x - 9
Distribution PrincipleMultiply everything in parenthesis by number next
to parenthesis FIRST THING
Get them off the busSo they can play football
EXAMPLE 3( 6x ndash 3)
3
18x - 9
6xndash 3
Example 2(10x ndash 3) = 6x + 2
Get the Teams off the Bus2(10x ndash 3) = 6x + 2
Line up the TeamsPenalty for off-sides ndash must change signs
Huddle up 14x = 14
Man on Man Defense 14x = 14 14 14
X = 1
Line of Scrimmage
20x ndash 6 = 6x + 8
20x ndash 6x = 8 + 6
Equation Solving
Think Football ndash Letters Vs Numbers
le rsaquo or lsaquo or geIf the equation has an Inequality sign follow the steps for solving equations with = signs
Play FootballLetters Vrs Numbers
Last Play of the Game
If you have to MULTIPLY or DIVIDE by a negative number Be sure to FLIP the Inequality
NOTICEUGLY NUMBERS WORK THE SAME AS PRETTY NUMBERS
Irsquom not afraid of
Fractions Have
Calculator Will
Calculate
UGLY Numbers Work the Same as PRETTY Numbers
If you can solve 2X + 10 = 40
Then you can solve 5x + 193 = 456
And you can solve
5x + ⅝ = ⅓
Play FootballLetters Vrs Numbers
Clue Words for writing equations from word problems
+
Word Clue
Plusadded to
the sum of increasing by
more than
Word Sentence
1 plus 56 is added to a numberThe sum of 5 and a numberA number is increased by 1015 is more than a number
Algebraic
1+56+x
5+x
x+10
x+15
Addition
Clue Words for writing equations from word problems
AlgebraicWord Clue
Minussubtracted fromthe difference ofdecreased byLessless than
Word Sentence
6 minus 57 subtracted from a numberThe difference of a number and 10A number is decreased by 205 less a number6 less than a number
6-5x-7
x-10
x-205-xx-6
Subtraction __
Clue Words for writing equations from word problems
Word Clue Word Sentence Algebraic
TimesProduct
DoubledTwiceOf (fractions and percents)
7 times a numberProduct of 8 and a numberA number doubledTwice a number12 a number55 of a number
7x = 7x8x
2x2x = 2x12x055x
Multiplication
Word Clue Word Sentence Algebraic
divide
Quotient
divided by
The quotient of a number and 710 divided by a number
xdivide7
10dividex
The first number written before the clue word will be the numerator
Clue Words for writing equations from word problems
Division
1 Consistent ndash one or many solutions2 Inconsistent ndash No solution
1 Independent ndash Only one solution2 Dependent ndash Has infinitely many
solutions
Slope Y ndash Int Graph Type of Systems
of Solutions
Same Same Same line ConsistentDependent
Infinitely many
Same Different Parallel Inconsistent 0
Different DifferentSame
Intersects ConsistentIndependent
1
Consistent and Inconsistent Systems
Slope ndash Intercept Form
y = mx + b
Slope- directions
RiseRun
Y Intercept ndash where to start
Itrsquos a line address
If the slope is a whole number put it on a stick m = 2 slope is 21
To Graph
y = 2X + 1 Starts at 1
Riserun = 21
Directions are up 2 over 1
Example 1 Example 2y = -3X+ 0
y = -3X
Starts at 0
riserun = 3-1
Directions are up 3 over -1
Thanks to httpwwwmathsisfuncomequation_of_linehtml
Linear Equations Standard Form ax + by = cSolving for y Just 3 easy stepsGreenisms Math Terms
1Move x term(Change sides Change signs) AddSubtract x term 2Give x side a hug Parenthesis ( )3Divide by number next to the y Coefficient
Example Solve for Y2x ndash 7y = 12Just 3 easy steps1 -7y = 12 ndash 2x (Move x term (Change sides Change signs)2 -7y = (12-2x) (Give it a hug)3 y = (12-2x) -7 (Divide by number next to the y)
Now you are ready to enter it into the calculator and graph it
WATCH YOUR SIGNS
Example Solve for Y2x ndash 7y = 12
Just 3 easy steps1 -7y = 12 ndash 2x X is offside Penalty change signs2 -7y = (12-2x) Huddle up ( )3 y = (12-2x) -7 Man on man defense
WATCH YOUR SIGNS
Linear Equations Standard Form ax + by = cSolving for y Itrsquos a Football Game
Y VS Everybody Else
Follow football rules
Play FootballY vs everybody else
Now you are ready to enter it into the calculator and graph it
l lines
Same Slope
Slopes are Negative Reciprocal
(Flip amp Change Sign)
PARALLEL LINES
y2 ndash y1
x2 ndash x1
or
y = mx + b
slope
Find Equation of the Line y = mx + b
I need slope (m) amp the y-intercept
(b)
MY ANSWER
y = x +
To find m ndash Solve the equation for y and use mor use the y2 ndash y1
x2 ndash x1 formula
To find b - Plug x y and m into the line equation and solve for b
(Just follow the Rules)
Base xsup2Exponent
Like Bases Exponent Example
Multiply Add
Power up Multiply
Divide Subtract s
Add No change 3asup2 + 5asup2 = 8asup2 asup2 + = asup2 +
Subtract No change 10asup2 - 4asup2 = 6asup2
5a
4
15 9 3a g y
6 4 5
3 7 4a x ka x k
5a
5a
5a
Exponents
=
(xsup2asup3g)(xasup2gsup3) = (xsup3 g )
( gsup3 y) sup3 =
asup3k xsup3
Negative ExponentsJust switch places and make exponent positive
5x5
1x
Example Switch Simplify
5 4 3
2 2 4x g ka x k
4 2
2 5 4 3g x
a x k k
4
2 3 7g
a x k
=
Xsup2
X X∙ = Xsup2 X
X times X is X squared
X
QuadraticsMultiplying Binomials ndash Draw the Face
(x + 8) (x ndash 6)
Multiply Watch your Signs
xsup2 +8x -6x -48
Simplify (Combine Like Terms eat the buggers)xsup2 + 2x -48
Check This First
The Math ldquoFrdquo Word ldquoFactoringrdquo
1 Is there a common factor (number or letter)
NO
Proceed to question 2
Yes
Proceed with CGF
4xsup3 2 2 x x x
4x ( 2xy + xsup2 -3 )
Example
8xsup2y + 4xsup3 - 12x
Factor each term8xsup2y 2 2 2 x x y 12x 2 2 3 x
Circle common terms
8xsup2y 2 2 2 x x y 4xsup3 2 2 x x x 12x 2 2 3 x
Multiply circled numbersThatrsquos your Common Factor
Multiply leftovers put in ( )
2 Perfect squares on end amp 3 terms
xsup2 - 10x + 25
YES SPLIT IT NICE
NO proceed to question 3
xsup2 - 10x + 25Put out baggies to hold the answer (parenthesis)
( x - 5 ) ( x - 5 )
Split the first term nice
Place in baggies
Sign between is same as middle term
Split the second term nice
Place in baggies
ANSWER
(x -5)sup2
3 Perfect squares on end amp 2 terms
YesSame as question 2Except the signsare +-
Example
xsup2 - 64
NO proceed to question 4
( x - 8 ) ( x + 8)
Answer looks like this
4 No perfect squares on end 3 terms amp starts with xsup2
NO proceed to question 5
YES its quadratics in the morning
xsup2 - 10x + 24MA
Multiply to +24 Add to -10
38 11
-3 -8 -11
-2 -12 -14
-6-4 -10 Found it
Put in the parenthesis
(x-6) (x-4)
All Done
2 Search and Seizure (quadratics in the morning)(See question 4)
1 Steal the ldquoardquo and give it to the last term (Multiply)1
5 Is there a number in front of the xsup2 amp does it have 3 terms
Yes Jail BreakNO
Then it is Prime
(canrsquot factor)
Example 2xsup2 + 7x + 3
xsup2 + 7x + 6 (23)
( x + 6 ) ( x + 1 )
3 Arrested and Caught - Divide last terms by ldquoardquo
( x + 62 ) ( x + 12 )
4 Beat it Down - (reduce fractions)
( x + 3 ) ( x + 12 )
5 Parole (kick denominator to the front)(x + 3 ) ( 2 x + 1 )
All Done
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Check This First
Is there a common factor (number or letter)
Greatest Common Factor (GCF)
1 Factor each term2 Circle common terms3 Multiply common terms4 Write it down5 Put out baggie for leftover6 Multiply leftovers for each
term7 Put in baggie ( )
Example
8xsup2y + 4xsup3 - 12x
4x ( 2xy + xsup2 - 3 )
Is there a Square on each End
Perfect Squares
1 Put out Parenthesis2 Split FIRST and LAST numbers nice3 Put in Signs
3 Different Kinds
A xsup2 - 16x + 64
(xndash 8) (x ndash 8)
B xsup2 + 18x + 81
(x + 9) (x + 9)
C xsup2 - 36
(x + 6) (x ndash 6)
Is there a number in front of the xsup2 and does it have 3 terms
Jail Break
1Steal the ldquoardquo and give it to the last term (multiply)2Search and Seizure (Quad in AM)3Arrested and Caught (divide by ldquoardquo)4Beat Down (reduce fractions)5Parole (kick denominator to the front)6Check it out (FACE it)
Example
2xsup2 + 7x + 3
1xsup2 + 7x + 62(x + 6) (x + 1)3(x + 62) (x + 12)4(x + 3) (x +12)5(x + 3) (2x + 1)
No perfect squares and 3 terms and starts with xsup2
Quadratics in the Morning (AM)
1 Make a factor tree2 Multiply to last number add
to middle number3 Put out baggies (parenthesis)4 Split first term nice5 Drop in factors from tree
Example
xsup2 + 12x + 32
(x + 8) (x + 4)
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Solve with Graphing Calculator
1) Solve each equation for y (3 easy steps)
2) Use y= button and enter each equation
3) Use graph to eyeball answerOr
4) Use to find where y1 and y2 are equal
5) Be sure answer is in (x y) form
Y=
Systems of EquationsTo Solve by Graphing
1) Make an x y Chart
2) Select any x solve for y x 2x ndash 4 y 0 2(0) ndash 4 -4 1 2(1) ndash 4 -2
3) Then graph the two points4) Do for both equations5) The answer is where they
cross6) Be sure answer is in (x y)
form
2nd TABLE
bull y + 2x = 9
bull y = 9 ndash 2xbull 3(9 - 2x) ndash 2x = 11
bull 27-6x ndash 2x = 11bull 27 ndash 8x = 11bull - 8x = -16bull x = 2bull y = 9 ndash 2 (2)bull y = 5bull (2 5)
Systems of EquationsSolve by Substitution
(box amp shove)
1) Solve one equation for x or y (change sides change signs)
2) Box it3) Rewrite other equation
and shove box in 4) Solve for surviving letter
1) Distribute2) Combine like terms3) Solve
4) Send it back to box5) Solve for other letter6) Answer in (xy) form
3y -2x = 11y + 2x = 9
Systems of EquationsSolve by Elimination
1) Look for opposite signs
2) Multiply to create opposites
3) Add old and new equation together
4) Solve for surviving letter
5) Plug back into either equation (pick the easy one)
6) Solve for other letter
7) Answer in (xy) format
bull 2x ndash y = 93x + 4y = -14
bull 4(2x ndash y = 9)
bull 8x ndash 4y = 36bull 3x + 4y = -14bull 11x = 22bull x = 2
bull 2(2) ndash y = 9bull -y = 5bull y = -5
bull (2 -5)
2x ndash y = 93x + 4y = -14
Distribution PrincipleMultiply everything in parenthesis by number next
to parenthesis FIRST THING
THINK BIG MOUTH SHOUTING ldquoDO MEDO ME FIRSTrdquo
EXAMPLE 3( 6x ndash 3)
3 (6x ndash 3) = 18x - 9
Distribution PrincipleMultiply everything in parenthesis by number next
to parenthesis FIRST THING
Get them off the busSo they can play football
EXAMPLE 3( 6x ndash 3)
3
18x - 9
6xndash 3
Example 2(10x ndash 3) = 6x + 2
Get the Teams off the Bus2(10x ndash 3) = 6x + 2
Line up the TeamsPenalty for off-sides ndash must change signs
Huddle up 14x = 14
Man on Man Defense 14x = 14 14 14
X = 1
Line of Scrimmage
20x ndash 6 = 6x + 8
20x ndash 6x = 8 + 6
Equation Solving
Think Football ndash Letters Vs Numbers
le rsaquo or lsaquo or geIf the equation has an Inequality sign follow the steps for solving equations with = signs
Play FootballLetters Vrs Numbers
Last Play of the Game
If you have to MULTIPLY or DIVIDE by a negative number Be sure to FLIP the Inequality
NOTICEUGLY NUMBERS WORK THE SAME AS PRETTY NUMBERS
Irsquom not afraid of
Fractions Have
Calculator Will
Calculate
UGLY Numbers Work the Same as PRETTY Numbers
If you can solve 2X + 10 = 40
Then you can solve 5x + 193 = 456
And you can solve
5x + ⅝ = ⅓
Play FootballLetters Vrs Numbers
Clue Words for writing equations from word problems
+
Word Clue
Plusadded to
the sum of increasing by
more than
Word Sentence
1 plus 56 is added to a numberThe sum of 5 and a numberA number is increased by 1015 is more than a number
Algebraic
1+56+x
5+x
x+10
x+15
Addition
Clue Words for writing equations from word problems
AlgebraicWord Clue
Minussubtracted fromthe difference ofdecreased byLessless than
Word Sentence
6 minus 57 subtracted from a numberThe difference of a number and 10A number is decreased by 205 less a number6 less than a number
6-5x-7
x-10
x-205-xx-6
Subtraction __
Clue Words for writing equations from word problems
Word Clue Word Sentence Algebraic
TimesProduct
DoubledTwiceOf (fractions and percents)
7 times a numberProduct of 8 and a numberA number doubledTwice a number12 a number55 of a number
7x = 7x8x
2x2x = 2x12x055x
Multiplication
Word Clue Word Sentence Algebraic
divide
Quotient
divided by
The quotient of a number and 710 divided by a number
xdivide7
10dividex
The first number written before the clue word will be the numerator
Clue Words for writing equations from word problems
Division
1 Consistent ndash one or many solutions2 Inconsistent ndash No solution
1 Independent ndash Only one solution2 Dependent ndash Has infinitely many
solutions
Slope Y ndash Int Graph Type of Systems
of Solutions
Same Same Same line ConsistentDependent
Infinitely many
Same Different Parallel Inconsistent 0
Different DifferentSame
Intersects ConsistentIndependent
1
Consistent and Inconsistent Systems
Slope ndash Intercept Form
y = mx + b
Slope- directions
RiseRun
Y Intercept ndash where to start
Itrsquos a line address
If the slope is a whole number put it on a stick m = 2 slope is 21
To Graph
y = 2X + 1 Starts at 1
Riserun = 21
Directions are up 2 over 1
Example 1 Example 2y = -3X+ 0
y = -3X
Starts at 0
riserun = 3-1
Directions are up 3 over -1
Thanks to httpwwwmathsisfuncomequation_of_linehtml
Linear Equations Standard Form ax + by = cSolving for y Just 3 easy stepsGreenisms Math Terms
1Move x term(Change sides Change signs) AddSubtract x term 2Give x side a hug Parenthesis ( )3Divide by number next to the y Coefficient
Example Solve for Y2x ndash 7y = 12Just 3 easy steps1 -7y = 12 ndash 2x (Move x term (Change sides Change signs)2 -7y = (12-2x) (Give it a hug)3 y = (12-2x) -7 (Divide by number next to the y)
Now you are ready to enter it into the calculator and graph it
WATCH YOUR SIGNS
Example Solve for Y2x ndash 7y = 12
Just 3 easy steps1 -7y = 12 ndash 2x X is offside Penalty change signs2 -7y = (12-2x) Huddle up ( )3 y = (12-2x) -7 Man on man defense
WATCH YOUR SIGNS
Linear Equations Standard Form ax + by = cSolving for y Itrsquos a Football Game
Y VS Everybody Else
Follow football rules
Play FootballY vs everybody else
Now you are ready to enter it into the calculator and graph it
l lines
Same Slope
Slopes are Negative Reciprocal
(Flip amp Change Sign)
PARALLEL LINES
y2 ndash y1
x2 ndash x1
or
y = mx + b
slope
Find Equation of the Line y = mx + b
I need slope (m) amp the y-intercept
(b)
MY ANSWER
y = x +
To find m ndash Solve the equation for y and use mor use the y2 ndash y1
x2 ndash x1 formula
To find b - Plug x y and m into the line equation and solve for b
(Just follow the Rules)
Base xsup2Exponent
Like Bases Exponent Example
Multiply Add
Power up Multiply
Divide Subtract s
Add No change 3asup2 + 5asup2 = 8asup2 asup2 + = asup2 +
Subtract No change 10asup2 - 4asup2 = 6asup2
5a
4
15 9 3a g y
6 4 5
3 7 4a x ka x k
5a
5a
5a
Exponents
=
(xsup2asup3g)(xasup2gsup3) = (xsup3 g )
( gsup3 y) sup3 =
asup3k xsup3
Negative ExponentsJust switch places and make exponent positive
5x5
1x
Example Switch Simplify
5 4 3
2 2 4x g ka x k
4 2
2 5 4 3g x
a x k k
4
2 3 7g
a x k
=
Xsup2
X X∙ = Xsup2 X
X times X is X squared
X
QuadraticsMultiplying Binomials ndash Draw the Face
(x + 8) (x ndash 6)
Multiply Watch your Signs
xsup2 +8x -6x -48
Simplify (Combine Like Terms eat the buggers)xsup2 + 2x -48
Check This First
The Math ldquoFrdquo Word ldquoFactoringrdquo
1 Is there a common factor (number or letter)
NO
Proceed to question 2
Yes
Proceed with CGF
4xsup3 2 2 x x x
4x ( 2xy + xsup2 -3 )
Example
8xsup2y + 4xsup3 - 12x
Factor each term8xsup2y 2 2 2 x x y 12x 2 2 3 x
Circle common terms
8xsup2y 2 2 2 x x y 4xsup3 2 2 x x x 12x 2 2 3 x
Multiply circled numbersThatrsquos your Common Factor
Multiply leftovers put in ( )
2 Perfect squares on end amp 3 terms
xsup2 - 10x + 25
YES SPLIT IT NICE
NO proceed to question 3
xsup2 - 10x + 25Put out baggies to hold the answer (parenthesis)
( x - 5 ) ( x - 5 )
Split the first term nice
Place in baggies
Sign between is same as middle term
Split the second term nice
Place in baggies
ANSWER
(x -5)sup2
3 Perfect squares on end amp 2 terms
YesSame as question 2Except the signsare +-
Example
xsup2 - 64
NO proceed to question 4
( x - 8 ) ( x + 8)
Answer looks like this
4 No perfect squares on end 3 terms amp starts with xsup2
NO proceed to question 5
YES its quadratics in the morning
xsup2 - 10x + 24MA
Multiply to +24 Add to -10
38 11
-3 -8 -11
-2 -12 -14
-6-4 -10 Found it
Put in the parenthesis
(x-6) (x-4)
All Done
2 Search and Seizure (quadratics in the morning)(See question 4)
1 Steal the ldquoardquo and give it to the last term (Multiply)1
5 Is there a number in front of the xsup2 amp does it have 3 terms
Yes Jail BreakNO
Then it is Prime
(canrsquot factor)
Example 2xsup2 + 7x + 3
xsup2 + 7x + 6 (23)
( x + 6 ) ( x + 1 )
3 Arrested and Caught - Divide last terms by ldquoardquo
( x + 62 ) ( x + 12 )
4 Beat it Down - (reduce fractions)
( x + 3 ) ( x + 12 )
5 Parole (kick denominator to the front)(x + 3 ) ( 2 x + 1 )
All Done
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Check This First
Is there a common factor (number or letter)
Greatest Common Factor (GCF)
1 Factor each term2 Circle common terms3 Multiply common terms4 Write it down5 Put out baggie for leftover6 Multiply leftovers for each
term7 Put in baggie ( )
Example
8xsup2y + 4xsup3 - 12x
4x ( 2xy + xsup2 - 3 )
Is there a Square on each End
Perfect Squares
1 Put out Parenthesis2 Split FIRST and LAST numbers nice3 Put in Signs
3 Different Kinds
A xsup2 - 16x + 64
(xndash 8) (x ndash 8)
B xsup2 + 18x + 81
(x + 9) (x + 9)
C xsup2 - 36
(x + 6) (x ndash 6)
Is there a number in front of the xsup2 and does it have 3 terms
Jail Break
1Steal the ldquoardquo and give it to the last term (multiply)2Search and Seizure (Quad in AM)3Arrested and Caught (divide by ldquoardquo)4Beat Down (reduce fractions)5Parole (kick denominator to the front)6Check it out (FACE it)
Example
2xsup2 + 7x + 3
1xsup2 + 7x + 62(x + 6) (x + 1)3(x + 62) (x + 12)4(x + 3) (x +12)5(x + 3) (2x + 1)
No perfect squares and 3 terms and starts with xsup2
Quadratics in the Morning (AM)
1 Make a factor tree2 Multiply to last number add
to middle number3 Put out baggies (parenthesis)4 Split first term nice5 Drop in factors from tree
Example
xsup2 + 12x + 32
(x + 8) (x + 4)
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Solve with Graphing Calculator
1) Solve each equation for y (3 easy steps)
2) Use y= button and enter each equation
3) Use graph to eyeball answerOr
4) Use to find where y1 and y2 are equal
5) Be sure answer is in (x y) form
Y=
Systems of EquationsTo Solve by Graphing
1) Make an x y Chart
2) Select any x solve for y x 2x ndash 4 y 0 2(0) ndash 4 -4 1 2(1) ndash 4 -2
3) Then graph the two points4) Do for both equations5) The answer is where they
cross6) Be sure answer is in (x y)
form
2nd TABLE
bull y + 2x = 9
bull y = 9 ndash 2xbull 3(9 - 2x) ndash 2x = 11
bull 27-6x ndash 2x = 11bull 27 ndash 8x = 11bull - 8x = -16bull x = 2bull y = 9 ndash 2 (2)bull y = 5bull (2 5)
Systems of EquationsSolve by Substitution
(box amp shove)
1) Solve one equation for x or y (change sides change signs)
2) Box it3) Rewrite other equation
and shove box in 4) Solve for surviving letter
1) Distribute2) Combine like terms3) Solve
4) Send it back to box5) Solve for other letter6) Answer in (xy) form
3y -2x = 11y + 2x = 9
Systems of EquationsSolve by Elimination
1) Look for opposite signs
2) Multiply to create opposites
3) Add old and new equation together
4) Solve for surviving letter
5) Plug back into either equation (pick the easy one)
6) Solve for other letter
7) Answer in (xy) format
bull 2x ndash y = 93x + 4y = -14
bull 4(2x ndash y = 9)
bull 8x ndash 4y = 36bull 3x + 4y = -14bull 11x = 22bull x = 2
bull 2(2) ndash y = 9bull -y = 5bull y = -5
bull (2 -5)
2x ndash y = 93x + 4y = -14
Distribution PrincipleMultiply everything in parenthesis by number next
to parenthesis FIRST THING
Get them off the busSo they can play football
EXAMPLE 3( 6x ndash 3)
3
18x - 9
6xndash 3
Example 2(10x ndash 3) = 6x + 2
Get the Teams off the Bus2(10x ndash 3) = 6x + 2
Line up the TeamsPenalty for off-sides ndash must change signs
Huddle up 14x = 14
Man on Man Defense 14x = 14 14 14
X = 1
Line of Scrimmage
20x ndash 6 = 6x + 8
20x ndash 6x = 8 + 6
Equation Solving
Think Football ndash Letters Vs Numbers
le rsaquo or lsaquo or geIf the equation has an Inequality sign follow the steps for solving equations with = signs
Play FootballLetters Vrs Numbers
Last Play of the Game
If you have to MULTIPLY or DIVIDE by a negative number Be sure to FLIP the Inequality
NOTICEUGLY NUMBERS WORK THE SAME AS PRETTY NUMBERS
Irsquom not afraid of
Fractions Have
Calculator Will
Calculate
UGLY Numbers Work the Same as PRETTY Numbers
If you can solve 2X + 10 = 40
Then you can solve 5x + 193 = 456
And you can solve
5x + ⅝ = ⅓
Play FootballLetters Vrs Numbers
Clue Words for writing equations from word problems
+
Word Clue
Plusadded to
the sum of increasing by
more than
Word Sentence
1 plus 56 is added to a numberThe sum of 5 and a numberA number is increased by 1015 is more than a number
Algebraic
1+56+x
5+x
x+10
x+15
Addition
Clue Words for writing equations from word problems
AlgebraicWord Clue
Minussubtracted fromthe difference ofdecreased byLessless than
Word Sentence
6 minus 57 subtracted from a numberThe difference of a number and 10A number is decreased by 205 less a number6 less than a number
6-5x-7
x-10
x-205-xx-6
Subtraction __
Clue Words for writing equations from word problems
Word Clue Word Sentence Algebraic
TimesProduct
DoubledTwiceOf (fractions and percents)
7 times a numberProduct of 8 and a numberA number doubledTwice a number12 a number55 of a number
7x = 7x8x
2x2x = 2x12x055x
Multiplication
Word Clue Word Sentence Algebraic
divide
Quotient
divided by
The quotient of a number and 710 divided by a number
xdivide7
10dividex
The first number written before the clue word will be the numerator
Clue Words for writing equations from word problems
Division
1 Consistent ndash one or many solutions2 Inconsistent ndash No solution
1 Independent ndash Only one solution2 Dependent ndash Has infinitely many
solutions
Slope Y ndash Int Graph Type of Systems
of Solutions
Same Same Same line ConsistentDependent
Infinitely many
Same Different Parallel Inconsistent 0
Different DifferentSame
Intersects ConsistentIndependent
1
Consistent and Inconsistent Systems
Slope ndash Intercept Form
y = mx + b
Slope- directions
RiseRun
Y Intercept ndash where to start
Itrsquos a line address
If the slope is a whole number put it on a stick m = 2 slope is 21
To Graph
y = 2X + 1 Starts at 1
Riserun = 21
Directions are up 2 over 1
Example 1 Example 2y = -3X+ 0
y = -3X
Starts at 0
riserun = 3-1
Directions are up 3 over -1
Thanks to httpwwwmathsisfuncomequation_of_linehtml
Linear Equations Standard Form ax + by = cSolving for y Just 3 easy stepsGreenisms Math Terms
1Move x term(Change sides Change signs) AddSubtract x term 2Give x side a hug Parenthesis ( )3Divide by number next to the y Coefficient
Example Solve for Y2x ndash 7y = 12Just 3 easy steps1 -7y = 12 ndash 2x (Move x term (Change sides Change signs)2 -7y = (12-2x) (Give it a hug)3 y = (12-2x) -7 (Divide by number next to the y)
Now you are ready to enter it into the calculator and graph it
WATCH YOUR SIGNS
Example Solve for Y2x ndash 7y = 12
Just 3 easy steps1 -7y = 12 ndash 2x X is offside Penalty change signs2 -7y = (12-2x) Huddle up ( )3 y = (12-2x) -7 Man on man defense
WATCH YOUR SIGNS
Linear Equations Standard Form ax + by = cSolving for y Itrsquos a Football Game
Y VS Everybody Else
Follow football rules
Play FootballY vs everybody else
Now you are ready to enter it into the calculator and graph it
l lines
Same Slope
Slopes are Negative Reciprocal
(Flip amp Change Sign)
PARALLEL LINES
y2 ndash y1
x2 ndash x1
or
y = mx + b
slope
Find Equation of the Line y = mx + b
I need slope (m) amp the y-intercept
(b)
MY ANSWER
y = x +
To find m ndash Solve the equation for y and use mor use the y2 ndash y1
x2 ndash x1 formula
To find b - Plug x y and m into the line equation and solve for b
(Just follow the Rules)
Base xsup2Exponent
Like Bases Exponent Example
Multiply Add
Power up Multiply
Divide Subtract s
Add No change 3asup2 + 5asup2 = 8asup2 asup2 + = asup2 +
Subtract No change 10asup2 - 4asup2 = 6asup2
5a
4
15 9 3a g y
6 4 5
3 7 4a x ka x k
5a
5a
5a
Exponents
=
(xsup2asup3g)(xasup2gsup3) = (xsup3 g )
( gsup3 y) sup3 =
asup3k xsup3
Negative ExponentsJust switch places and make exponent positive
5x5
1x
Example Switch Simplify
5 4 3
2 2 4x g ka x k
4 2
2 5 4 3g x
a x k k
4
2 3 7g
a x k
=
Xsup2
X X∙ = Xsup2 X
X times X is X squared
X
QuadraticsMultiplying Binomials ndash Draw the Face
(x + 8) (x ndash 6)
Multiply Watch your Signs
xsup2 +8x -6x -48
Simplify (Combine Like Terms eat the buggers)xsup2 + 2x -48
Check This First
The Math ldquoFrdquo Word ldquoFactoringrdquo
1 Is there a common factor (number or letter)
NO
Proceed to question 2
Yes
Proceed with CGF
4xsup3 2 2 x x x
4x ( 2xy + xsup2 -3 )
Example
8xsup2y + 4xsup3 - 12x
Factor each term8xsup2y 2 2 2 x x y 12x 2 2 3 x
Circle common terms
8xsup2y 2 2 2 x x y 4xsup3 2 2 x x x 12x 2 2 3 x
Multiply circled numbersThatrsquos your Common Factor
Multiply leftovers put in ( )
2 Perfect squares on end amp 3 terms
xsup2 - 10x + 25
YES SPLIT IT NICE
NO proceed to question 3
xsup2 - 10x + 25Put out baggies to hold the answer (parenthesis)
( x - 5 ) ( x - 5 )
Split the first term nice
Place in baggies
Sign between is same as middle term
Split the second term nice
Place in baggies
ANSWER
(x -5)sup2
3 Perfect squares on end amp 2 terms
YesSame as question 2Except the signsare +-
Example
xsup2 - 64
NO proceed to question 4
( x - 8 ) ( x + 8)
Answer looks like this
4 No perfect squares on end 3 terms amp starts with xsup2
NO proceed to question 5
YES its quadratics in the morning
xsup2 - 10x + 24MA
Multiply to +24 Add to -10
38 11
-3 -8 -11
-2 -12 -14
-6-4 -10 Found it
Put in the parenthesis
(x-6) (x-4)
All Done
2 Search and Seizure (quadratics in the morning)(See question 4)
1 Steal the ldquoardquo and give it to the last term (Multiply)1
5 Is there a number in front of the xsup2 amp does it have 3 terms
Yes Jail BreakNO
Then it is Prime
(canrsquot factor)
Example 2xsup2 + 7x + 3
xsup2 + 7x + 6 (23)
( x + 6 ) ( x + 1 )
3 Arrested and Caught - Divide last terms by ldquoardquo
( x + 62 ) ( x + 12 )
4 Beat it Down - (reduce fractions)
( x + 3 ) ( x + 12 )
5 Parole (kick denominator to the front)(x + 3 ) ( 2 x + 1 )
All Done
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Check This First
Is there a common factor (number or letter)
Greatest Common Factor (GCF)
1 Factor each term2 Circle common terms3 Multiply common terms4 Write it down5 Put out baggie for leftover6 Multiply leftovers for each
term7 Put in baggie ( )
Example
8xsup2y + 4xsup3 - 12x
4x ( 2xy + xsup2 - 3 )
Is there a Square on each End
Perfect Squares
1 Put out Parenthesis2 Split FIRST and LAST numbers nice3 Put in Signs
3 Different Kinds
A xsup2 - 16x + 64
(xndash 8) (x ndash 8)
B xsup2 + 18x + 81
(x + 9) (x + 9)
C xsup2 - 36
(x + 6) (x ndash 6)
Is there a number in front of the xsup2 and does it have 3 terms
Jail Break
1Steal the ldquoardquo and give it to the last term (multiply)2Search and Seizure (Quad in AM)3Arrested and Caught (divide by ldquoardquo)4Beat Down (reduce fractions)5Parole (kick denominator to the front)6Check it out (FACE it)
Example
2xsup2 + 7x + 3
1xsup2 + 7x + 62(x + 6) (x + 1)3(x + 62) (x + 12)4(x + 3) (x +12)5(x + 3) (2x + 1)
No perfect squares and 3 terms and starts with xsup2
Quadratics in the Morning (AM)
1 Make a factor tree2 Multiply to last number add
to middle number3 Put out baggies (parenthesis)4 Split first term nice5 Drop in factors from tree
Example
xsup2 + 12x + 32
(x + 8) (x + 4)
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Solve with Graphing Calculator
1) Solve each equation for y (3 easy steps)
2) Use y= button and enter each equation
3) Use graph to eyeball answerOr
4) Use to find where y1 and y2 are equal
5) Be sure answer is in (x y) form
Y=
Systems of EquationsTo Solve by Graphing
1) Make an x y Chart
2) Select any x solve for y x 2x ndash 4 y 0 2(0) ndash 4 -4 1 2(1) ndash 4 -2
3) Then graph the two points4) Do for both equations5) The answer is where they
cross6) Be sure answer is in (x y)
form
2nd TABLE
bull y + 2x = 9
bull y = 9 ndash 2xbull 3(9 - 2x) ndash 2x = 11
bull 27-6x ndash 2x = 11bull 27 ndash 8x = 11bull - 8x = -16bull x = 2bull y = 9 ndash 2 (2)bull y = 5bull (2 5)
Systems of EquationsSolve by Substitution
(box amp shove)
1) Solve one equation for x or y (change sides change signs)
2) Box it3) Rewrite other equation
and shove box in 4) Solve for surviving letter
1) Distribute2) Combine like terms3) Solve
4) Send it back to box5) Solve for other letter6) Answer in (xy) form
3y -2x = 11y + 2x = 9
Systems of EquationsSolve by Elimination
1) Look for opposite signs
2) Multiply to create opposites
3) Add old and new equation together
4) Solve for surviving letter
5) Plug back into either equation (pick the easy one)
6) Solve for other letter
7) Answer in (xy) format
bull 2x ndash y = 93x + 4y = -14
bull 4(2x ndash y = 9)
bull 8x ndash 4y = 36bull 3x + 4y = -14bull 11x = 22bull x = 2
bull 2(2) ndash y = 9bull -y = 5bull y = -5
bull (2 -5)
2x ndash y = 93x + 4y = -14
Example 2(10x ndash 3) = 6x + 2
Get the Teams off the Bus2(10x ndash 3) = 6x + 2
Line up the TeamsPenalty for off-sides ndash must change signs
Huddle up 14x = 14
Man on Man Defense 14x = 14 14 14
X = 1
Line of Scrimmage
20x ndash 6 = 6x + 8
20x ndash 6x = 8 + 6
Equation Solving
Think Football ndash Letters Vs Numbers
le rsaquo or lsaquo or geIf the equation has an Inequality sign follow the steps for solving equations with = signs
Play FootballLetters Vrs Numbers
Last Play of the Game
If you have to MULTIPLY or DIVIDE by a negative number Be sure to FLIP the Inequality
NOTICEUGLY NUMBERS WORK THE SAME AS PRETTY NUMBERS
Irsquom not afraid of
Fractions Have
Calculator Will
Calculate
UGLY Numbers Work the Same as PRETTY Numbers
If you can solve 2X + 10 = 40
Then you can solve 5x + 193 = 456
And you can solve
5x + ⅝ = ⅓
Play FootballLetters Vrs Numbers
Clue Words for writing equations from word problems
+
Word Clue
Plusadded to
the sum of increasing by
more than
Word Sentence
1 plus 56 is added to a numberThe sum of 5 and a numberA number is increased by 1015 is more than a number
Algebraic
1+56+x
5+x
x+10
x+15
Addition
Clue Words for writing equations from word problems
AlgebraicWord Clue
Minussubtracted fromthe difference ofdecreased byLessless than
Word Sentence
6 minus 57 subtracted from a numberThe difference of a number and 10A number is decreased by 205 less a number6 less than a number
6-5x-7
x-10
x-205-xx-6
Subtraction __
Clue Words for writing equations from word problems
Word Clue Word Sentence Algebraic
TimesProduct
DoubledTwiceOf (fractions and percents)
7 times a numberProduct of 8 and a numberA number doubledTwice a number12 a number55 of a number
7x = 7x8x
2x2x = 2x12x055x
Multiplication
Word Clue Word Sentence Algebraic
divide
Quotient
divided by
The quotient of a number and 710 divided by a number
xdivide7
10dividex
The first number written before the clue word will be the numerator
Clue Words for writing equations from word problems
Division
1 Consistent ndash one or many solutions2 Inconsistent ndash No solution
1 Independent ndash Only one solution2 Dependent ndash Has infinitely many
solutions
Slope Y ndash Int Graph Type of Systems
of Solutions
Same Same Same line ConsistentDependent
Infinitely many
Same Different Parallel Inconsistent 0
Different DifferentSame
Intersects ConsistentIndependent
1
Consistent and Inconsistent Systems
Slope ndash Intercept Form
y = mx + b
Slope- directions
RiseRun
Y Intercept ndash where to start
Itrsquos a line address
If the slope is a whole number put it on a stick m = 2 slope is 21
To Graph
y = 2X + 1 Starts at 1
Riserun = 21
Directions are up 2 over 1
Example 1 Example 2y = -3X+ 0
y = -3X
Starts at 0
riserun = 3-1
Directions are up 3 over -1
Thanks to httpwwwmathsisfuncomequation_of_linehtml
Linear Equations Standard Form ax + by = cSolving for y Just 3 easy stepsGreenisms Math Terms
1Move x term(Change sides Change signs) AddSubtract x term 2Give x side a hug Parenthesis ( )3Divide by number next to the y Coefficient
Example Solve for Y2x ndash 7y = 12Just 3 easy steps1 -7y = 12 ndash 2x (Move x term (Change sides Change signs)2 -7y = (12-2x) (Give it a hug)3 y = (12-2x) -7 (Divide by number next to the y)
Now you are ready to enter it into the calculator and graph it
WATCH YOUR SIGNS
Example Solve for Y2x ndash 7y = 12
Just 3 easy steps1 -7y = 12 ndash 2x X is offside Penalty change signs2 -7y = (12-2x) Huddle up ( )3 y = (12-2x) -7 Man on man defense
WATCH YOUR SIGNS
Linear Equations Standard Form ax + by = cSolving for y Itrsquos a Football Game
Y VS Everybody Else
Follow football rules
Play FootballY vs everybody else
Now you are ready to enter it into the calculator and graph it
l lines
Same Slope
Slopes are Negative Reciprocal
(Flip amp Change Sign)
PARALLEL LINES
y2 ndash y1
x2 ndash x1
or
y = mx + b
slope
Find Equation of the Line y = mx + b
I need slope (m) amp the y-intercept
(b)
MY ANSWER
y = x +
To find m ndash Solve the equation for y and use mor use the y2 ndash y1
x2 ndash x1 formula
To find b - Plug x y and m into the line equation and solve for b
(Just follow the Rules)
Base xsup2Exponent
Like Bases Exponent Example
Multiply Add
Power up Multiply
Divide Subtract s
Add No change 3asup2 + 5asup2 = 8asup2 asup2 + = asup2 +
Subtract No change 10asup2 - 4asup2 = 6asup2
5a
4
15 9 3a g y
6 4 5
3 7 4a x ka x k
5a
5a
5a
Exponents
=
(xsup2asup3g)(xasup2gsup3) = (xsup3 g )
( gsup3 y) sup3 =
asup3k xsup3
Negative ExponentsJust switch places and make exponent positive
5x5
1x
Example Switch Simplify
5 4 3
2 2 4x g ka x k
4 2
2 5 4 3g x
a x k k
4
2 3 7g
a x k
=
Xsup2
X X∙ = Xsup2 X
X times X is X squared
X
QuadraticsMultiplying Binomials ndash Draw the Face
(x + 8) (x ndash 6)
Multiply Watch your Signs
xsup2 +8x -6x -48
Simplify (Combine Like Terms eat the buggers)xsup2 + 2x -48
Check This First
The Math ldquoFrdquo Word ldquoFactoringrdquo
1 Is there a common factor (number or letter)
NO
Proceed to question 2
Yes
Proceed with CGF
4xsup3 2 2 x x x
4x ( 2xy + xsup2 -3 )
Example
8xsup2y + 4xsup3 - 12x
Factor each term8xsup2y 2 2 2 x x y 12x 2 2 3 x
Circle common terms
8xsup2y 2 2 2 x x y 4xsup3 2 2 x x x 12x 2 2 3 x
Multiply circled numbersThatrsquos your Common Factor
Multiply leftovers put in ( )
2 Perfect squares on end amp 3 terms
xsup2 - 10x + 25
YES SPLIT IT NICE
NO proceed to question 3
xsup2 - 10x + 25Put out baggies to hold the answer (parenthesis)
( x - 5 ) ( x - 5 )
Split the first term nice
Place in baggies
Sign between is same as middle term
Split the second term nice
Place in baggies
ANSWER
(x -5)sup2
3 Perfect squares on end amp 2 terms
YesSame as question 2Except the signsare +-
Example
xsup2 - 64
NO proceed to question 4
( x - 8 ) ( x + 8)
Answer looks like this
4 No perfect squares on end 3 terms amp starts with xsup2
NO proceed to question 5
YES its quadratics in the morning
xsup2 - 10x + 24MA
Multiply to +24 Add to -10
38 11
-3 -8 -11
-2 -12 -14
-6-4 -10 Found it
Put in the parenthesis
(x-6) (x-4)
All Done
2 Search and Seizure (quadratics in the morning)(See question 4)
1 Steal the ldquoardquo and give it to the last term (Multiply)1
5 Is there a number in front of the xsup2 amp does it have 3 terms
Yes Jail BreakNO
Then it is Prime
(canrsquot factor)
Example 2xsup2 + 7x + 3
xsup2 + 7x + 6 (23)
( x + 6 ) ( x + 1 )
3 Arrested and Caught - Divide last terms by ldquoardquo
( x + 62 ) ( x + 12 )
4 Beat it Down - (reduce fractions)
( x + 3 ) ( x + 12 )
5 Parole (kick denominator to the front)(x + 3 ) ( 2 x + 1 )
All Done
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Check This First
Is there a common factor (number or letter)
Greatest Common Factor (GCF)
1 Factor each term2 Circle common terms3 Multiply common terms4 Write it down5 Put out baggie for leftover6 Multiply leftovers for each
term7 Put in baggie ( )
Example
8xsup2y + 4xsup3 - 12x
4x ( 2xy + xsup2 - 3 )
Is there a Square on each End
Perfect Squares
1 Put out Parenthesis2 Split FIRST and LAST numbers nice3 Put in Signs
3 Different Kinds
A xsup2 - 16x + 64
(xndash 8) (x ndash 8)
B xsup2 + 18x + 81
(x + 9) (x + 9)
C xsup2 - 36
(x + 6) (x ndash 6)
Is there a number in front of the xsup2 and does it have 3 terms
Jail Break
1Steal the ldquoardquo and give it to the last term (multiply)2Search and Seizure (Quad in AM)3Arrested and Caught (divide by ldquoardquo)4Beat Down (reduce fractions)5Parole (kick denominator to the front)6Check it out (FACE it)
Example
2xsup2 + 7x + 3
1xsup2 + 7x + 62(x + 6) (x + 1)3(x + 62) (x + 12)4(x + 3) (x +12)5(x + 3) (2x + 1)
No perfect squares and 3 terms and starts with xsup2
Quadratics in the Morning (AM)
1 Make a factor tree2 Multiply to last number add
to middle number3 Put out baggies (parenthesis)4 Split first term nice5 Drop in factors from tree
Example
xsup2 + 12x + 32
(x + 8) (x + 4)
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Solve with Graphing Calculator
1) Solve each equation for y (3 easy steps)
2) Use y= button and enter each equation
3) Use graph to eyeball answerOr
4) Use to find where y1 and y2 are equal
5) Be sure answer is in (x y) form
Y=
Systems of EquationsTo Solve by Graphing
1) Make an x y Chart
2) Select any x solve for y x 2x ndash 4 y 0 2(0) ndash 4 -4 1 2(1) ndash 4 -2
3) Then graph the two points4) Do for both equations5) The answer is where they
cross6) Be sure answer is in (x y)
form
2nd TABLE
bull y + 2x = 9
bull y = 9 ndash 2xbull 3(9 - 2x) ndash 2x = 11
bull 27-6x ndash 2x = 11bull 27 ndash 8x = 11bull - 8x = -16bull x = 2bull y = 9 ndash 2 (2)bull y = 5bull (2 5)
Systems of EquationsSolve by Substitution
(box amp shove)
1) Solve one equation for x or y (change sides change signs)
2) Box it3) Rewrite other equation
and shove box in 4) Solve for surviving letter
1) Distribute2) Combine like terms3) Solve
4) Send it back to box5) Solve for other letter6) Answer in (xy) form
3y -2x = 11y + 2x = 9
Systems of EquationsSolve by Elimination
1) Look for opposite signs
2) Multiply to create opposites
3) Add old and new equation together
4) Solve for surviving letter
5) Plug back into either equation (pick the easy one)
6) Solve for other letter
7) Answer in (xy) format
bull 2x ndash y = 93x + 4y = -14
bull 4(2x ndash y = 9)
bull 8x ndash 4y = 36bull 3x + 4y = -14bull 11x = 22bull x = 2
bull 2(2) ndash y = 9bull -y = 5bull y = -5
bull (2 -5)
2x ndash y = 93x + 4y = -14
le rsaquo or lsaquo or geIf the equation has an Inequality sign follow the steps for solving equations with = signs
Play FootballLetters Vrs Numbers
Last Play of the Game
If you have to MULTIPLY or DIVIDE by a negative number Be sure to FLIP the Inequality
NOTICEUGLY NUMBERS WORK THE SAME AS PRETTY NUMBERS
Irsquom not afraid of
Fractions Have
Calculator Will
Calculate
UGLY Numbers Work the Same as PRETTY Numbers
If you can solve 2X + 10 = 40
Then you can solve 5x + 193 = 456
And you can solve
5x + ⅝ = ⅓
Play FootballLetters Vrs Numbers
Clue Words for writing equations from word problems
+
Word Clue
Plusadded to
the sum of increasing by
more than
Word Sentence
1 plus 56 is added to a numberThe sum of 5 and a numberA number is increased by 1015 is more than a number
Algebraic
1+56+x
5+x
x+10
x+15
Addition
Clue Words for writing equations from word problems
AlgebraicWord Clue
Minussubtracted fromthe difference ofdecreased byLessless than
Word Sentence
6 minus 57 subtracted from a numberThe difference of a number and 10A number is decreased by 205 less a number6 less than a number
6-5x-7
x-10
x-205-xx-6
Subtraction __
Clue Words for writing equations from word problems
Word Clue Word Sentence Algebraic
TimesProduct
DoubledTwiceOf (fractions and percents)
7 times a numberProduct of 8 and a numberA number doubledTwice a number12 a number55 of a number
7x = 7x8x
2x2x = 2x12x055x
Multiplication
Word Clue Word Sentence Algebraic
divide
Quotient
divided by
The quotient of a number and 710 divided by a number
xdivide7
10dividex
The first number written before the clue word will be the numerator
Clue Words for writing equations from word problems
Division
1 Consistent ndash one or many solutions2 Inconsistent ndash No solution
1 Independent ndash Only one solution2 Dependent ndash Has infinitely many
solutions
Slope Y ndash Int Graph Type of Systems
of Solutions
Same Same Same line ConsistentDependent
Infinitely many
Same Different Parallel Inconsistent 0
Different DifferentSame
Intersects ConsistentIndependent
1
Consistent and Inconsistent Systems
Slope ndash Intercept Form
y = mx + b
Slope- directions
RiseRun
Y Intercept ndash where to start
Itrsquos a line address
If the slope is a whole number put it on a stick m = 2 slope is 21
To Graph
y = 2X + 1 Starts at 1
Riserun = 21
Directions are up 2 over 1
Example 1 Example 2y = -3X+ 0
y = -3X
Starts at 0
riserun = 3-1
Directions are up 3 over -1
Thanks to httpwwwmathsisfuncomequation_of_linehtml
Linear Equations Standard Form ax + by = cSolving for y Just 3 easy stepsGreenisms Math Terms
1Move x term(Change sides Change signs) AddSubtract x term 2Give x side a hug Parenthesis ( )3Divide by number next to the y Coefficient
Example Solve for Y2x ndash 7y = 12Just 3 easy steps1 -7y = 12 ndash 2x (Move x term (Change sides Change signs)2 -7y = (12-2x) (Give it a hug)3 y = (12-2x) -7 (Divide by number next to the y)
Now you are ready to enter it into the calculator and graph it
WATCH YOUR SIGNS
Example Solve for Y2x ndash 7y = 12
Just 3 easy steps1 -7y = 12 ndash 2x X is offside Penalty change signs2 -7y = (12-2x) Huddle up ( )3 y = (12-2x) -7 Man on man defense
WATCH YOUR SIGNS
Linear Equations Standard Form ax + by = cSolving for y Itrsquos a Football Game
Y VS Everybody Else
Follow football rules
Play FootballY vs everybody else
Now you are ready to enter it into the calculator and graph it
l lines
Same Slope
Slopes are Negative Reciprocal
(Flip amp Change Sign)
PARALLEL LINES
y2 ndash y1
x2 ndash x1
or
y = mx + b
slope
Find Equation of the Line y = mx + b
I need slope (m) amp the y-intercept
(b)
MY ANSWER
y = x +
To find m ndash Solve the equation for y and use mor use the y2 ndash y1
x2 ndash x1 formula
To find b - Plug x y and m into the line equation and solve for b
(Just follow the Rules)
Base xsup2Exponent
Like Bases Exponent Example
Multiply Add
Power up Multiply
Divide Subtract s
Add No change 3asup2 + 5asup2 = 8asup2 asup2 + = asup2 +
Subtract No change 10asup2 - 4asup2 = 6asup2
5a
4
15 9 3a g y
6 4 5
3 7 4a x ka x k
5a
5a
5a
Exponents
=
(xsup2asup3g)(xasup2gsup3) = (xsup3 g )
( gsup3 y) sup3 =
asup3k xsup3
Negative ExponentsJust switch places and make exponent positive
5x5
1x
Example Switch Simplify
5 4 3
2 2 4x g ka x k
4 2
2 5 4 3g x
a x k k
4
2 3 7g
a x k
=
Xsup2
X X∙ = Xsup2 X
X times X is X squared
X
QuadraticsMultiplying Binomials ndash Draw the Face
(x + 8) (x ndash 6)
Multiply Watch your Signs
xsup2 +8x -6x -48
Simplify (Combine Like Terms eat the buggers)xsup2 + 2x -48
Check This First
The Math ldquoFrdquo Word ldquoFactoringrdquo
1 Is there a common factor (number or letter)
NO
Proceed to question 2
Yes
Proceed with CGF
4xsup3 2 2 x x x
4x ( 2xy + xsup2 -3 )
Example
8xsup2y + 4xsup3 - 12x
Factor each term8xsup2y 2 2 2 x x y 12x 2 2 3 x
Circle common terms
8xsup2y 2 2 2 x x y 4xsup3 2 2 x x x 12x 2 2 3 x
Multiply circled numbersThatrsquos your Common Factor
Multiply leftovers put in ( )
2 Perfect squares on end amp 3 terms
xsup2 - 10x + 25
YES SPLIT IT NICE
NO proceed to question 3
xsup2 - 10x + 25Put out baggies to hold the answer (parenthesis)
( x - 5 ) ( x - 5 )
Split the first term nice
Place in baggies
Sign between is same as middle term
Split the second term nice
Place in baggies
ANSWER
(x -5)sup2
3 Perfect squares on end amp 2 terms
YesSame as question 2Except the signsare +-
Example
xsup2 - 64
NO proceed to question 4
( x - 8 ) ( x + 8)
Answer looks like this
4 No perfect squares on end 3 terms amp starts with xsup2
NO proceed to question 5
YES its quadratics in the morning
xsup2 - 10x + 24MA
Multiply to +24 Add to -10
38 11
-3 -8 -11
-2 -12 -14
-6-4 -10 Found it
Put in the parenthesis
(x-6) (x-4)
All Done
2 Search and Seizure (quadratics in the morning)(See question 4)
1 Steal the ldquoardquo and give it to the last term (Multiply)1
5 Is there a number in front of the xsup2 amp does it have 3 terms
Yes Jail BreakNO
Then it is Prime
(canrsquot factor)
Example 2xsup2 + 7x + 3
xsup2 + 7x + 6 (23)
( x + 6 ) ( x + 1 )
3 Arrested and Caught - Divide last terms by ldquoardquo
( x + 62 ) ( x + 12 )
4 Beat it Down - (reduce fractions)
( x + 3 ) ( x + 12 )
5 Parole (kick denominator to the front)(x + 3 ) ( 2 x + 1 )
All Done
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Check This First
Is there a common factor (number or letter)
Greatest Common Factor (GCF)
1 Factor each term2 Circle common terms3 Multiply common terms4 Write it down5 Put out baggie for leftover6 Multiply leftovers for each
term7 Put in baggie ( )
Example
8xsup2y + 4xsup3 - 12x
4x ( 2xy + xsup2 - 3 )
Is there a Square on each End
Perfect Squares
1 Put out Parenthesis2 Split FIRST and LAST numbers nice3 Put in Signs
3 Different Kinds
A xsup2 - 16x + 64
(xndash 8) (x ndash 8)
B xsup2 + 18x + 81
(x + 9) (x + 9)
C xsup2 - 36
(x + 6) (x ndash 6)
Is there a number in front of the xsup2 and does it have 3 terms
Jail Break
1Steal the ldquoardquo and give it to the last term (multiply)2Search and Seizure (Quad in AM)3Arrested and Caught (divide by ldquoardquo)4Beat Down (reduce fractions)5Parole (kick denominator to the front)6Check it out (FACE it)
Example
2xsup2 + 7x + 3
1xsup2 + 7x + 62(x + 6) (x + 1)3(x + 62) (x + 12)4(x + 3) (x +12)5(x + 3) (2x + 1)
No perfect squares and 3 terms and starts with xsup2
Quadratics in the Morning (AM)
1 Make a factor tree2 Multiply to last number add
to middle number3 Put out baggies (parenthesis)4 Split first term nice5 Drop in factors from tree
Example
xsup2 + 12x + 32
(x + 8) (x + 4)
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Solve with Graphing Calculator
1) Solve each equation for y (3 easy steps)
2) Use y= button and enter each equation
3) Use graph to eyeball answerOr
4) Use to find where y1 and y2 are equal
5) Be sure answer is in (x y) form
Y=
Systems of EquationsTo Solve by Graphing
1) Make an x y Chart
2) Select any x solve for y x 2x ndash 4 y 0 2(0) ndash 4 -4 1 2(1) ndash 4 -2
3) Then graph the two points4) Do for both equations5) The answer is where they
cross6) Be sure answer is in (x y)
form
2nd TABLE
bull y + 2x = 9
bull y = 9 ndash 2xbull 3(9 - 2x) ndash 2x = 11
bull 27-6x ndash 2x = 11bull 27 ndash 8x = 11bull - 8x = -16bull x = 2bull y = 9 ndash 2 (2)bull y = 5bull (2 5)
Systems of EquationsSolve by Substitution
(box amp shove)
1) Solve one equation for x or y (change sides change signs)
2) Box it3) Rewrite other equation
and shove box in 4) Solve for surviving letter
1) Distribute2) Combine like terms3) Solve
4) Send it back to box5) Solve for other letter6) Answer in (xy) form
3y -2x = 11y + 2x = 9
Systems of EquationsSolve by Elimination
1) Look for opposite signs
2) Multiply to create opposites
3) Add old and new equation together
4) Solve for surviving letter
5) Plug back into either equation (pick the easy one)
6) Solve for other letter
7) Answer in (xy) format
bull 2x ndash y = 93x + 4y = -14
bull 4(2x ndash y = 9)
bull 8x ndash 4y = 36bull 3x + 4y = -14bull 11x = 22bull x = 2
bull 2(2) ndash y = 9bull -y = 5bull y = -5
bull (2 -5)
2x ndash y = 93x + 4y = -14
Irsquom not afraid of
Fractions Have
Calculator Will
Calculate
UGLY Numbers Work the Same as PRETTY Numbers
If you can solve 2X + 10 = 40
Then you can solve 5x + 193 = 456
And you can solve
5x + ⅝ = ⅓
Play FootballLetters Vrs Numbers
Clue Words for writing equations from word problems
+
Word Clue
Plusadded to
the sum of increasing by
more than
Word Sentence
1 plus 56 is added to a numberThe sum of 5 and a numberA number is increased by 1015 is more than a number
Algebraic
1+56+x
5+x
x+10
x+15
Addition
Clue Words for writing equations from word problems
AlgebraicWord Clue
Minussubtracted fromthe difference ofdecreased byLessless than
Word Sentence
6 minus 57 subtracted from a numberThe difference of a number and 10A number is decreased by 205 less a number6 less than a number
6-5x-7
x-10
x-205-xx-6
Subtraction __
Clue Words for writing equations from word problems
Word Clue Word Sentence Algebraic
TimesProduct
DoubledTwiceOf (fractions and percents)
7 times a numberProduct of 8 and a numberA number doubledTwice a number12 a number55 of a number
7x = 7x8x
2x2x = 2x12x055x
Multiplication
Word Clue Word Sentence Algebraic
divide
Quotient
divided by
The quotient of a number and 710 divided by a number
xdivide7
10dividex
The first number written before the clue word will be the numerator
Clue Words for writing equations from word problems
Division
1 Consistent ndash one or many solutions2 Inconsistent ndash No solution
1 Independent ndash Only one solution2 Dependent ndash Has infinitely many
solutions
Slope Y ndash Int Graph Type of Systems
of Solutions
Same Same Same line ConsistentDependent
Infinitely many
Same Different Parallel Inconsistent 0
Different DifferentSame
Intersects ConsistentIndependent
1
Consistent and Inconsistent Systems
Slope ndash Intercept Form
y = mx + b
Slope- directions
RiseRun
Y Intercept ndash where to start
Itrsquos a line address
If the slope is a whole number put it on a stick m = 2 slope is 21
To Graph
y = 2X + 1 Starts at 1
Riserun = 21
Directions are up 2 over 1
Example 1 Example 2y = -3X+ 0
y = -3X
Starts at 0
riserun = 3-1
Directions are up 3 over -1
Thanks to httpwwwmathsisfuncomequation_of_linehtml
Linear Equations Standard Form ax + by = cSolving for y Just 3 easy stepsGreenisms Math Terms
1Move x term(Change sides Change signs) AddSubtract x term 2Give x side a hug Parenthesis ( )3Divide by number next to the y Coefficient
Example Solve for Y2x ndash 7y = 12Just 3 easy steps1 -7y = 12 ndash 2x (Move x term (Change sides Change signs)2 -7y = (12-2x) (Give it a hug)3 y = (12-2x) -7 (Divide by number next to the y)
Now you are ready to enter it into the calculator and graph it
WATCH YOUR SIGNS
Example Solve for Y2x ndash 7y = 12
Just 3 easy steps1 -7y = 12 ndash 2x X is offside Penalty change signs2 -7y = (12-2x) Huddle up ( )3 y = (12-2x) -7 Man on man defense
WATCH YOUR SIGNS
Linear Equations Standard Form ax + by = cSolving for y Itrsquos a Football Game
Y VS Everybody Else
Follow football rules
Play FootballY vs everybody else
Now you are ready to enter it into the calculator and graph it
l lines
Same Slope
Slopes are Negative Reciprocal
(Flip amp Change Sign)
PARALLEL LINES
y2 ndash y1
x2 ndash x1
or
y = mx + b
slope
Find Equation of the Line y = mx + b
I need slope (m) amp the y-intercept
(b)
MY ANSWER
y = x +
To find m ndash Solve the equation for y and use mor use the y2 ndash y1
x2 ndash x1 formula
To find b - Plug x y and m into the line equation and solve for b
(Just follow the Rules)
Base xsup2Exponent
Like Bases Exponent Example
Multiply Add
Power up Multiply
Divide Subtract s
Add No change 3asup2 + 5asup2 = 8asup2 asup2 + = asup2 +
Subtract No change 10asup2 - 4asup2 = 6asup2
5a
4
15 9 3a g y
6 4 5
3 7 4a x ka x k
5a
5a
5a
Exponents
=
(xsup2asup3g)(xasup2gsup3) = (xsup3 g )
( gsup3 y) sup3 =
asup3k xsup3
Negative ExponentsJust switch places and make exponent positive
5x5
1x
Example Switch Simplify
5 4 3
2 2 4x g ka x k
4 2
2 5 4 3g x
a x k k
4
2 3 7g
a x k
=
Xsup2
X X∙ = Xsup2 X
X times X is X squared
X
QuadraticsMultiplying Binomials ndash Draw the Face
(x + 8) (x ndash 6)
Multiply Watch your Signs
xsup2 +8x -6x -48
Simplify (Combine Like Terms eat the buggers)xsup2 + 2x -48
Check This First
The Math ldquoFrdquo Word ldquoFactoringrdquo
1 Is there a common factor (number or letter)
NO
Proceed to question 2
Yes
Proceed with CGF
4xsup3 2 2 x x x
4x ( 2xy + xsup2 -3 )
Example
8xsup2y + 4xsup3 - 12x
Factor each term8xsup2y 2 2 2 x x y 12x 2 2 3 x
Circle common terms
8xsup2y 2 2 2 x x y 4xsup3 2 2 x x x 12x 2 2 3 x
Multiply circled numbersThatrsquos your Common Factor
Multiply leftovers put in ( )
2 Perfect squares on end amp 3 terms
xsup2 - 10x + 25
YES SPLIT IT NICE
NO proceed to question 3
xsup2 - 10x + 25Put out baggies to hold the answer (parenthesis)
( x - 5 ) ( x - 5 )
Split the first term nice
Place in baggies
Sign between is same as middle term
Split the second term nice
Place in baggies
ANSWER
(x -5)sup2
3 Perfect squares on end amp 2 terms
YesSame as question 2Except the signsare +-
Example
xsup2 - 64
NO proceed to question 4
( x - 8 ) ( x + 8)
Answer looks like this
4 No perfect squares on end 3 terms amp starts with xsup2
NO proceed to question 5
YES its quadratics in the morning
xsup2 - 10x + 24MA
Multiply to +24 Add to -10
38 11
-3 -8 -11
-2 -12 -14
-6-4 -10 Found it
Put in the parenthesis
(x-6) (x-4)
All Done
2 Search and Seizure (quadratics in the morning)(See question 4)
1 Steal the ldquoardquo and give it to the last term (Multiply)1
5 Is there a number in front of the xsup2 amp does it have 3 terms
Yes Jail BreakNO
Then it is Prime
(canrsquot factor)
Example 2xsup2 + 7x + 3
xsup2 + 7x + 6 (23)
( x + 6 ) ( x + 1 )
3 Arrested and Caught - Divide last terms by ldquoardquo
( x + 62 ) ( x + 12 )
4 Beat it Down - (reduce fractions)
( x + 3 ) ( x + 12 )
5 Parole (kick denominator to the front)(x + 3 ) ( 2 x + 1 )
All Done
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Check This First
Is there a common factor (number or letter)
Greatest Common Factor (GCF)
1 Factor each term2 Circle common terms3 Multiply common terms4 Write it down5 Put out baggie for leftover6 Multiply leftovers for each
term7 Put in baggie ( )
Example
8xsup2y + 4xsup3 - 12x
4x ( 2xy + xsup2 - 3 )
Is there a Square on each End
Perfect Squares
1 Put out Parenthesis2 Split FIRST and LAST numbers nice3 Put in Signs
3 Different Kinds
A xsup2 - 16x + 64
(xndash 8) (x ndash 8)
B xsup2 + 18x + 81
(x + 9) (x + 9)
C xsup2 - 36
(x + 6) (x ndash 6)
Is there a number in front of the xsup2 and does it have 3 terms
Jail Break
1Steal the ldquoardquo and give it to the last term (multiply)2Search and Seizure (Quad in AM)3Arrested and Caught (divide by ldquoardquo)4Beat Down (reduce fractions)5Parole (kick denominator to the front)6Check it out (FACE it)
Example
2xsup2 + 7x + 3
1xsup2 + 7x + 62(x + 6) (x + 1)3(x + 62) (x + 12)4(x + 3) (x +12)5(x + 3) (2x + 1)
No perfect squares and 3 terms and starts with xsup2
Quadratics in the Morning (AM)
1 Make a factor tree2 Multiply to last number add
to middle number3 Put out baggies (parenthesis)4 Split first term nice5 Drop in factors from tree
Example
xsup2 + 12x + 32
(x + 8) (x + 4)
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Solve with Graphing Calculator
1) Solve each equation for y (3 easy steps)
2) Use y= button and enter each equation
3) Use graph to eyeball answerOr
4) Use to find where y1 and y2 are equal
5) Be sure answer is in (x y) form
Y=
Systems of EquationsTo Solve by Graphing
1) Make an x y Chart
2) Select any x solve for y x 2x ndash 4 y 0 2(0) ndash 4 -4 1 2(1) ndash 4 -2
3) Then graph the two points4) Do for both equations5) The answer is where they
cross6) Be sure answer is in (x y)
form
2nd TABLE
bull y + 2x = 9
bull y = 9 ndash 2xbull 3(9 - 2x) ndash 2x = 11
bull 27-6x ndash 2x = 11bull 27 ndash 8x = 11bull - 8x = -16bull x = 2bull y = 9 ndash 2 (2)bull y = 5bull (2 5)
Systems of EquationsSolve by Substitution
(box amp shove)
1) Solve one equation for x or y (change sides change signs)
2) Box it3) Rewrite other equation
and shove box in 4) Solve for surviving letter
1) Distribute2) Combine like terms3) Solve
4) Send it back to box5) Solve for other letter6) Answer in (xy) form
3y -2x = 11y + 2x = 9
Systems of EquationsSolve by Elimination
1) Look for opposite signs
2) Multiply to create opposites
3) Add old and new equation together
4) Solve for surviving letter
5) Plug back into either equation (pick the easy one)
6) Solve for other letter
7) Answer in (xy) format
bull 2x ndash y = 93x + 4y = -14
bull 4(2x ndash y = 9)
bull 8x ndash 4y = 36bull 3x + 4y = -14bull 11x = 22bull x = 2
bull 2(2) ndash y = 9bull -y = 5bull y = -5
bull (2 -5)
2x ndash y = 93x + 4y = -14
Clue Words for writing equations from word problems
+
Word Clue
Plusadded to
the sum of increasing by
more than
Word Sentence
1 plus 56 is added to a numberThe sum of 5 and a numberA number is increased by 1015 is more than a number
Algebraic
1+56+x
5+x
x+10
x+15
Addition
Clue Words for writing equations from word problems
AlgebraicWord Clue
Minussubtracted fromthe difference ofdecreased byLessless than
Word Sentence
6 minus 57 subtracted from a numberThe difference of a number and 10A number is decreased by 205 less a number6 less than a number
6-5x-7
x-10
x-205-xx-6
Subtraction __
Clue Words for writing equations from word problems
Word Clue Word Sentence Algebraic
TimesProduct
DoubledTwiceOf (fractions and percents)
7 times a numberProduct of 8 and a numberA number doubledTwice a number12 a number55 of a number
7x = 7x8x
2x2x = 2x12x055x
Multiplication
Word Clue Word Sentence Algebraic
divide
Quotient
divided by
The quotient of a number and 710 divided by a number
xdivide7
10dividex
The first number written before the clue word will be the numerator
Clue Words for writing equations from word problems
Division
1 Consistent ndash one or many solutions2 Inconsistent ndash No solution
1 Independent ndash Only one solution2 Dependent ndash Has infinitely many
solutions
Slope Y ndash Int Graph Type of Systems
of Solutions
Same Same Same line ConsistentDependent
Infinitely many
Same Different Parallel Inconsistent 0
Different DifferentSame
Intersects ConsistentIndependent
1
Consistent and Inconsistent Systems
Slope ndash Intercept Form
y = mx + b
Slope- directions
RiseRun
Y Intercept ndash where to start
Itrsquos a line address
If the slope is a whole number put it on a stick m = 2 slope is 21
To Graph
y = 2X + 1 Starts at 1
Riserun = 21
Directions are up 2 over 1
Example 1 Example 2y = -3X+ 0
y = -3X
Starts at 0
riserun = 3-1
Directions are up 3 over -1
Thanks to httpwwwmathsisfuncomequation_of_linehtml
Linear Equations Standard Form ax + by = cSolving for y Just 3 easy stepsGreenisms Math Terms
1Move x term(Change sides Change signs) AddSubtract x term 2Give x side a hug Parenthesis ( )3Divide by number next to the y Coefficient
Example Solve for Y2x ndash 7y = 12Just 3 easy steps1 -7y = 12 ndash 2x (Move x term (Change sides Change signs)2 -7y = (12-2x) (Give it a hug)3 y = (12-2x) -7 (Divide by number next to the y)
Now you are ready to enter it into the calculator and graph it
WATCH YOUR SIGNS
Example Solve for Y2x ndash 7y = 12
Just 3 easy steps1 -7y = 12 ndash 2x X is offside Penalty change signs2 -7y = (12-2x) Huddle up ( )3 y = (12-2x) -7 Man on man defense
WATCH YOUR SIGNS
Linear Equations Standard Form ax + by = cSolving for y Itrsquos a Football Game
Y VS Everybody Else
Follow football rules
Play FootballY vs everybody else
Now you are ready to enter it into the calculator and graph it
l lines
Same Slope
Slopes are Negative Reciprocal
(Flip amp Change Sign)
PARALLEL LINES
y2 ndash y1
x2 ndash x1
or
y = mx + b
slope
Find Equation of the Line y = mx + b
I need slope (m) amp the y-intercept
(b)
MY ANSWER
y = x +
To find m ndash Solve the equation for y and use mor use the y2 ndash y1
x2 ndash x1 formula
To find b - Plug x y and m into the line equation and solve for b
(Just follow the Rules)
Base xsup2Exponent
Like Bases Exponent Example
Multiply Add
Power up Multiply
Divide Subtract s
Add No change 3asup2 + 5asup2 = 8asup2 asup2 + = asup2 +
Subtract No change 10asup2 - 4asup2 = 6asup2
5a
4
15 9 3a g y
6 4 5
3 7 4a x ka x k
5a
5a
5a
Exponents
=
(xsup2asup3g)(xasup2gsup3) = (xsup3 g )
( gsup3 y) sup3 =
asup3k xsup3
Negative ExponentsJust switch places and make exponent positive
5x5
1x
Example Switch Simplify
5 4 3
2 2 4x g ka x k
4 2
2 5 4 3g x
a x k k
4
2 3 7g
a x k
=
Xsup2
X X∙ = Xsup2 X
X times X is X squared
X
QuadraticsMultiplying Binomials ndash Draw the Face
(x + 8) (x ndash 6)
Multiply Watch your Signs
xsup2 +8x -6x -48
Simplify (Combine Like Terms eat the buggers)xsup2 + 2x -48
Check This First
The Math ldquoFrdquo Word ldquoFactoringrdquo
1 Is there a common factor (number or letter)
NO
Proceed to question 2
Yes
Proceed with CGF
4xsup3 2 2 x x x
4x ( 2xy + xsup2 -3 )
Example
8xsup2y + 4xsup3 - 12x
Factor each term8xsup2y 2 2 2 x x y 12x 2 2 3 x
Circle common terms
8xsup2y 2 2 2 x x y 4xsup3 2 2 x x x 12x 2 2 3 x
Multiply circled numbersThatrsquos your Common Factor
Multiply leftovers put in ( )
2 Perfect squares on end amp 3 terms
xsup2 - 10x + 25
YES SPLIT IT NICE
NO proceed to question 3
xsup2 - 10x + 25Put out baggies to hold the answer (parenthesis)
( x - 5 ) ( x - 5 )
Split the first term nice
Place in baggies
Sign between is same as middle term
Split the second term nice
Place in baggies
ANSWER
(x -5)sup2
3 Perfect squares on end amp 2 terms
YesSame as question 2Except the signsare +-
Example
xsup2 - 64
NO proceed to question 4
( x - 8 ) ( x + 8)
Answer looks like this
4 No perfect squares on end 3 terms amp starts with xsup2
NO proceed to question 5
YES its quadratics in the morning
xsup2 - 10x + 24MA
Multiply to +24 Add to -10
38 11
-3 -8 -11
-2 -12 -14
-6-4 -10 Found it
Put in the parenthesis
(x-6) (x-4)
All Done
2 Search and Seizure (quadratics in the morning)(See question 4)
1 Steal the ldquoardquo and give it to the last term (Multiply)1
5 Is there a number in front of the xsup2 amp does it have 3 terms
Yes Jail BreakNO
Then it is Prime
(canrsquot factor)
Example 2xsup2 + 7x + 3
xsup2 + 7x + 6 (23)
( x + 6 ) ( x + 1 )
3 Arrested and Caught - Divide last terms by ldquoardquo
( x + 62 ) ( x + 12 )
4 Beat it Down - (reduce fractions)
( x + 3 ) ( x + 12 )
5 Parole (kick denominator to the front)(x + 3 ) ( 2 x + 1 )
All Done
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Check This First
Is there a common factor (number or letter)
Greatest Common Factor (GCF)
1 Factor each term2 Circle common terms3 Multiply common terms4 Write it down5 Put out baggie for leftover6 Multiply leftovers for each
term7 Put in baggie ( )
Example
8xsup2y + 4xsup3 - 12x
4x ( 2xy + xsup2 - 3 )
Is there a Square on each End
Perfect Squares
1 Put out Parenthesis2 Split FIRST and LAST numbers nice3 Put in Signs
3 Different Kinds
A xsup2 - 16x + 64
(xndash 8) (x ndash 8)
B xsup2 + 18x + 81
(x + 9) (x + 9)
C xsup2 - 36
(x + 6) (x ndash 6)
Is there a number in front of the xsup2 and does it have 3 terms
Jail Break
1Steal the ldquoardquo and give it to the last term (multiply)2Search and Seizure (Quad in AM)3Arrested and Caught (divide by ldquoardquo)4Beat Down (reduce fractions)5Parole (kick denominator to the front)6Check it out (FACE it)
Example
2xsup2 + 7x + 3
1xsup2 + 7x + 62(x + 6) (x + 1)3(x + 62) (x + 12)4(x + 3) (x +12)5(x + 3) (2x + 1)
No perfect squares and 3 terms and starts with xsup2
Quadratics in the Morning (AM)
1 Make a factor tree2 Multiply to last number add
to middle number3 Put out baggies (parenthesis)4 Split first term nice5 Drop in factors from tree
Example
xsup2 + 12x + 32
(x + 8) (x + 4)
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Solve with Graphing Calculator
1) Solve each equation for y (3 easy steps)
2) Use y= button and enter each equation
3) Use graph to eyeball answerOr
4) Use to find where y1 and y2 are equal
5) Be sure answer is in (x y) form
Y=
Systems of EquationsTo Solve by Graphing
1) Make an x y Chart
2) Select any x solve for y x 2x ndash 4 y 0 2(0) ndash 4 -4 1 2(1) ndash 4 -2
3) Then graph the two points4) Do for both equations5) The answer is where they
cross6) Be sure answer is in (x y)
form
2nd TABLE
bull y + 2x = 9
bull y = 9 ndash 2xbull 3(9 - 2x) ndash 2x = 11
bull 27-6x ndash 2x = 11bull 27 ndash 8x = 11bull - 8x = -16bull x = 2bull y = 9 ndash 2 (2)bull y = 5bull (2 5)
Systems of EquationsSolve by Substitution
(box amp shove)
1) Solve one equation for x or y (change sides change signs)
2) Box it3) Rewrite other equation
and shove box in 4) Solve for surviving letter
1) Distribute2) Combine like terms3) Solve
4) Send it back to box5) Solve for other letter6) Answer in (xy) form
3y -2x = 11y + 2x = 9
Systems of EquationsSolve by Elimination
1) Look for opposite signs
2) Multiply to create opposites
3) Add old and new equation together
4) Solve for surviving letter
5) Plug back into either equation (pick the easy one)
6) Solve for other letter
7) Answer in (xy) format
bull 2x ndash y = 93x + 4y = -14
bull 4(2x ndash y = 9)
bull 8x ndash 4y = 36bull 3x + 4y = -14bull 11x = 22bull x = 2
bull 2(2) ndash y = 9bull -y = 5bull y = -5
bull (2 -5)
2x ndash y = 93x + 4y = -14
Clue Words for writing equations from word problems
AlgebraicWord Clue
Minussubtracted fromthe difference ofdecreased byLessless than
Word Sentence
6 minus 57 subtracted from a numberThe difference of a number and 10A number is decreased by 205 less a number6 less than a number
6-5x-7
x-10
x-205-xx-6
Subtraction __
Clue Words for writing equations from word problems
Word Clue Word Sentence Algebraic
TimesProduct
DoubledTwiceOf (fractions and percents)
7 times a numberProduct of 8 and a numberA number doubledTwice a number12 a number55 of a number
7x = 7x8x
2x2x = 2x12x055x
Multiplication
Word Clue Word Sentence Algebraic
divide
Quotient
divided by
The quotient of a number and 710 divided by a number
xdivide7
10dividex
The first number written before the clue word will be the numerator
Clue Words for writing equations from word problems
Division
1 Consistent ndash one or many solutions2 Inconsistent ndash No solution
1 Independent ndash Only one solution2 Dependent ndash Has infinitely many
solutions
Slope Y ndash Int Graph Type of Systems
of Solutions
Same Same Same line ConsistentDependent
Infinitely many
Same Different Parallel Inconsistent 0
Different DifferentSame
Intersects ConsistentIndependent
1
Consistent and Inconsistent Systems
Slope ndash Intercept Form
y = mx + b
Slope- directions
RiseRun
Y Intercept ndash where to start
Itrsquos a line address
If the slope is a whole number put it on a stick m = 2 slope is 21
To Graph
y = 2X + 1 Starts at 1
Riserun = 21
Directions are up 2 over 1
Example 1 Example 2y = -3X+ 0
y = -3X
Starts at 0
riserun = 3-1
Directions are up 3 over -1
Thanks to httpwwwmathsisfuncomequation_of_linehtml
Linear Equations Standard Form ax + by = cSolving for y Just 3 easy stepsGreenisms Math Terms
1Move x term(Change sides Change signs) AddSubtract x term 2Give x side a hug Parenthesis ( )3Divide by number next to the y Coefficient
Example Solve for Y2x ndash 7y = 12Just 3 easy steps1 -7y = 12 ndash 2x (Move x term (Change sides Change signs)2 -7y = (12-2x) (Give it a hug)3 y = (12-2x) -7 (Divide by number next to the y)
Now you are ready to enter it into the calculator and graph it
WATCH YOUR SIGNS
Example Solve for Y2x ndash 7y = 12
Just 3 easy steps1 -7y = 12 ndash 2x X is offside Penalty change signs2 -7y = (12-2x) Huddle up ( )3 y = (12-2x) -7 Man on man defense
WATCH YOUR SIGNS
Linear Equations Standard Form ax + by = cSolving for y Itrsquos a Football Game
Y VS Everybody Else
Follow football rules
Play FootballY vs everybody else
Now you are ready to enter it into the calculator and graph it
l lines
Same Slope
Slopes are Negative Reciprocal
(Flip amp Change Sign)
PARALLEL LINES
y2 ndash y1
x2 ndash x1
or
y = mx + b
slope
Find Equation of the Line y = mx + b
I need slope (m) amp the y-intercept
(b)
MY ANSWER
y = x +
To find m ndash Solve the equation for y and use mor use the y2 ndash y1
x2 ndash x1 formula
To find b - Plug x y and m into the line equation and solve for b
(Just follow the Rules)
Base xsup2Exponent
Like Bases Exponent Example
Multiply Add
Power up Multiply
Divide Subtract s
Add No change 3asup2 + 5asup2 = 8asup2 asup2 + = asup2 +
Subtract No change 10asup2 - 4asup2 = 6asup2
5a
4
15 9 3a g y
6 4 5
3 7 4a x ka x k
5a
5a
5a
Exponents
=
(xsup2asup3g)(xasup2gsup3) = (xsup3 g )
( gsup3 y) sup3 =
asup3k xsup3
Negative ExponentsJust switch places and make exponent positive
5x5
1x
Example Switch Simplify
5 4 3
2 2 4x g ka x k
4 2
2 5 4 3g x
a x k k
4
2 3 7g
a x k
=
Xsup2
X X∙ = Xsup2 X
X times X is X squared
X
QuadraticsMultiplying Binomials ndash Draw the Face
(x + 8) (x ndash 6)
Multiply Watch your Signs
xsup2 +8x -6x -48
Simplify (Combine Like Terms eat the buggers)xsup2 + 2x -48
Check This First
The Math ldquoFrdquo Word ldquoFactoringrdquo
1 Is there a common factor (number or letter)
NO
Proceed to question 2
Yes
Proceed with CGF
4xsup3 2 2 x x x
4x ( 2xy + xsup2 -3 )
Example
8xsup2y + 4xsup3 - 12x
Factor each term8xsup2y 2 2 2 x x y 12x 2 2 3 x
Circle common terms
8xsup2y 2 2 2 x x y 4xsup3 2 2 x x x 12x 2 2 3 x
Multiply circled numbersThatrsquos your Common Factor
Multiply leftovers put in ( )
2 Perfect squares on end amp 3 terms
xsup2 - 10x + 25
YES SPLIT IT NICE
NO proceed to question 3
xsup2 - 10x + 25Put out baggies to hold the answer (parenthesis)
( x - 5 ) ( x - 5 )
Split the first term nice
Place in baggies
Sign between is same as middle term
Split the second term nice
Place in baggies
ANSWER
(x -5)sup2
3 Perfect squares on end amp 2 terms
YesSame as question 2Except the signsare +-
Example
xsup2 - 64
NO proceed to question 4
( x - 8 ) ( x + 8)
Answer looks like this
4 No perfect squares on end 3 terms amp starts with xsup2
NO proceed to question 5
YES its quadratics in the morning
xsup2 - 10x + 24MA
Multiply to +24 Add to -10
38 11
-3 -8 -11
-2 -12 -14
-6-4 -10 Found it
Put in the parenthesis
(x-6) (x-4)
All Done
2 Search and Seizure (quadratics in the morning)(See question 4)
1 Steal the ldquoardquo and give it to the last term (Multiply)1
5 Is there a number in front of the xsup2 amp does it have 3 terms
Yes Jail BreakNO
Then it is Prime
(canrsquot factor)
Example 2xsup2 + 7x + 3
xsup2 + 7x + 6 (23)
( x + 6 ) ( x + 1 )
3 Arrested and Caught - Divide last terms by ldquoardquo
( x + 62 ) ( x + 12 )
4 Beat it Down - (reduce fractions)
( x + 3 ) ( x + 12 )
5 Parole (kick denominator to the front)(x + 3 ) ( 2 x + 1 )
All Done
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Check This First
Is there a common factor (number or letter)
Greatest Common Factor (GCF)
1 Factor each term2 Circle common terms3 Multiply common terms4 Write it down5 Put out baggie for leftover6 Multiply leftovers for each
term7 Put in baggie ( )
Example
8xsup2y + 4xsup3 - 12x
4x ( 2xy + xsup2 - 3 )
Is there a Square on each End
Perfect Squares
1 Put out Parenthesis2 Split FIRST and LAST numbers nice3 Put in Signs
3 Different Kinds
A xsup2 - 16x + 64
(xndash 8) (x ndash 8)
B xsup2 + 18x + 81
(x + 9) (x + 9)
C xsup2 - 36
(x + 6) (x ndash 6)
Is there a number in front of the xsup2 and does it have 3 terms
Jail Break
1Steal the ldquoardquo and give it to the last term (multiply)2Search and Seizure (Quad in AM)3Arrested and Caught (divide by ldquoardquo)4Beat Down (reduce fractions)5Parole (kick denominator to the front)6Check it out (FACE it)
Example
2xsup2 + 7x + 3
1xsup2 + 7x + 62(x + 6) (x + 1)3(x + 62) (x + 12)4(x + 3) (x +12)5(x + 3) (2x + 1)
No perfect squares and 3 terms and starts with xsup2
Quadratics in the Morning (AM)
1 Make a factor tree2 Multiply to last number add
to middle number3 Put out baggies (parenthesis)4 Split first term nice5 Drop in factors from tree
Example
xsup2 + 12x + 32
(x + 8) (x + 4)
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Solve with Graphing Calculator
1) Solve each equation for y (3 easy steps)
2) Use y= button and enter each equation
3) Use graph to eyeball answerOr
4) Use to find where y1 and y2 are equal
5) Be sure answer is in (x y) form
Y=
Systems of EquationsTo Solve by Graphing
1) Make an x y Chart
2) Select any x solve for y x 2x ndash 4 y 0 2(0) ndash 4 -4 1 2(1) ndash 4 -2
3) Then graph the two points4) Do for both equations5) The answer is where they
cross6) Be sure answer is in (x y)
form
2nd TABLE
bull y + 2x = 9
bull y = 9 ndash 2xbull 3(9 - 2x) ndash 2x = 11
bull 27-6x ndash 2x = 11bull 27 ndash 8x = 11bull - 8x = -16bull x = 2bull y = 9 ndash 2 (2)bull y = 5bull (2 5)
Systems of EquationsSolve by Substitution
(box amp shove)
1) Solve one equation for x or y (change sides change signs)
2) Box it3) Rewrite other equation
and shove box in 4) Solve for surviving letter
1) Distribute2) Combine like terms3) Solve
4) Send it back to box5) Solve for other letter6) Answer in (xy) form
3y -2x = 11y + 2x = 9
Systems of EquationsSolve by Elimination
1) Look for opposite signs
2) Multiply to create opposites
3) Add old and new equation together
4) Solve for surviving letter
5) Plug back into either equation (pick the easy one)
6) Solve for other letter
7) Answer in (xy) format
bull 2x ndash y = 93x + 4y = -14
bull 4(2x ndash y = 9)
bull 8x ndash 4y = 36bull 3x + 4y = -14bull 11x = 22bull x = 2
bull 2(2) ndash y = 9bull -y = 5bull y = -5
bull (2 -5)
2x ndash y = 93x + 4y = -14
Clue Words for writing equations from word problems
Word Clue Word Sentence Algebraic
TimesProduct
DoubledTwiceOf (fractions and percents)
7 times a numberProduct of 8 and a numberA number doubledTwice a number12 a number55 of a number
7x = 7x8x
2x2x = 2x12x055x
Multiplication
Word Clue Word Sentence Algebraic
divide
Quotient
divided by
The quotient of a number and 710 divided by a number
xdivide7
10dividex
The first number written before the clue word will be the numerator
Clue Words for writing equations from word problems
Division
1 Consistent ndash one or many solutions2 Inconsistent ndash No solution
1 Independent ndash Only one solution2 Dependent ndash Has infinitely many
solutions
Slope Y ndash Int Graph Type of Systems
of Solutions
Same Same Same line ConsistentDependent
Infinitely many
Same Different Parallel Inconsistent 0
Different DifferentSame
Intersects ConsistentIndependent
1
Consistent and Inconsistent Systems
Slope ndash Intercept Form
y = mx + b
Slope- directions
RiseRun
Y Intercept ndash where to start
Itrsquos a line address
If the slope is a whole number put it on a stick m = 2 slope is 21
To Graph
y = 2X + 1 Starts at 1
Riserun = 21
Directions are up 2 over 1
Example 1 Example 2y = -3X+ 0
y = -3X
Starts at 0
riserun = 3-1
Directions are up 3 over -1
Thanks to httpwwwmathsisfuncomequation_of_linehtml
Linear Equations Standard Form ax + by = cSolving for y Just 3 easy stepsGreenisms Math Terms
1Move x term(Change sides Change signs) AddSubtract x term 2Give x side a hug Parenthesis ( )3Divide by number next to the y Coefficient
Example Solve for Y2x ndash 7y = 12Just 3 easy steps1 -7y = 12 ndash 2x (Move x term (Change sides Change signs)2 -7y = (12-2x) (Give it a hug)3 y = (12-2x) -7 (Divide by number next to the y)
Now you are ready to enter it into the calculator and graph it
WATCH YOUR SIGNS
Example Solve for Y2x ndash 7y = 12
Just 3 easy steps1 -7y = 12 ndash 2x X is offside Penalty change signs2 -7y = (12-2x) Huddle up ( )3 y = (12-2x) -7 Man on man defense
WATCH YOUR SIGNS
Linear Equations Standard Form ax + by = cSolving for y Itrsquos a Football Game
Y VS Everybody Else
Follow football rules
Play FootballY vs everybody else
Now you are ready to enter it into the calculator and graph it
l lines
Same Slope
Slopes are Negative Reciprocal
(Flip amp Change Sign)
PARALLEL LINES
y2 ndash y1
x2 ndash x1
or
y = mx + b
slope
Find Equation of the Line y = mx + b
I need slope (m) amp the y-intercept
(b)
MY ANSWER
y = x +
To find m ndash Solve the equation for y and use mor use the y2 ndash y1
x2 ndash x1 formula
To find b - Plug x y and m into the line equation and solve for b
(Just follow the Rules)
Base xsup2Exponent
Like Bases Exponent Example
Multiply Add
Power up Multiply
Divide Subtract s
Add No change 3asup2 + 5asup2 = 8asup2 asup2 + = asup2 +
Subtract No change 10asup2 - 4asup2 = 6asup2
5a
4
15 9 3a g y
6 4 5
3 7 4a x ka x k
5a
5a
5a
Exponents
=
(xsup2asup3g)(xasup2gsup3) = (xsup3 g )
( gsup3 y) sup3 =
asup3k xsup3
Negative ExponentsJust switch places and make exponent positive
5x5
1x
Example Switch Simplify
5 4 3
2 2 4x g ka x k
4 2
2 5 4 3g x
a x k k
4
2 3 7g
a x k
=
Xsup2
X X∙ = Xsup2 X
X times X is X squared
X
QuadraticsMultiplying Binomials ndash Draw the Face
(x + 8) (x ndash 6)
Multiply Watch your Signs
xsup2 +8x -6x -48
Simplify (Combine Like Terms eat the buggers)xsup2 + 2x -48
Check This First
The Math ldquoFrdquo Word ldquoFactoringrdquo
1 Is there a common factor (number or letter)
NO
Proceed to question 2
Yes
Proceed with CGF
4xsup3 2 2 x x x
4x ( 2xy + xsup2 -3 )
Example
8xsup2y + 4xsup3 - 12x
Factor each term8xsup2y 2 2 2 x x y 12x 2 2 3 x
Circle common terms
8xsup2y 2 2 2 x x y 4xsup3 2 2 x x x 12x 2 2 3 x
Multiply circled numbersThatrsquos your Common Factor
Multiply leftovers put in ( )
2 Perfect squares on end amp 3 terms
xsup2 - 10x + 25
YES SPLIT IT NICE
NO proceed to question 3
xsup2 - 10x + 25Put out baggies to hold the answer (parenthesis)
( x - 5 ) ( x - 5 )
Split the first term nice
Place in baggies
Sign between is same as middle term
Split the second term nice
Place in baggies
ANSWER
(x -5)sup2
3 Perfect squares on end amp 2 terms
YesSame as question 2Except the signsare +-
Example
xsup2 - 64
NO proceed to question 4
( x - 8 ) ( x + 8)
Answer looks like this
4 No perfect squares on end 3 terms amp starts with xsup2
NO proceed to question 5
YES its quadratics in the morning
xsup2 - 10x + 24MA
Multiply to +24 Add to -10
38 11
-3 -8 -11
-2 -12 -14
-6-4 -10 Found it
Put in the parenthesis
(x-6) (x-4)
All Done
2 Search and Seizure (quadratics in the morning)(See question 4)
1 Steal the ldquoardquo and give it to the last term (Multiply)1
5 Is there a number in front of the xsup2 amp does it have 3 terms
Yes Jail BreakNO
Then it is Prime
(canrsquot factor)
Example 2xsup2 + 7x + 3
xsup2 + 7x + 6 (23)
( x + 6 ) ( x + 1 )
3 Arrested and Caught - Divide last terms by ldquoardquo
( x + 62 ) ( x + 12 )
4 Beat it Down - (reduce fractions)
( x + 3 ) ( x + 12 )
5 Parole (kick denominator to the front)(x + 3 ) ( 2 x + 1 )
All Done
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Check This First
Is there a common factor (number or letter)
Greatest Common Factor (GCF)
1 Factor each term2 Circle common terms3 Multiply common terms4 Write it down5 Put out baggie for leftover6 Multiply leftovers for each
term7 Put in baggie ( )
Example
8xsup2y + 4xsup3 - 12x
4x ( 2xy + xsup2 - 3 )
Is there a Square on each End
Perfect Squares
1 Put out Parenthesis2 Split FIRST and LAST numbers nice3 Put in Signs
3 Different Kinds
A xsup2 - 16x + 64
(xndash 8) (x ndash 8)
B xsup2 + 18x + 81
(x + 9) (x + 9)
C xsup2 - 36
(x + 6) (x ndash 6)
Is there a number in front of the xsup2 and does it have 3 terms
Jail Break
1Steal the ldquoardquo and give it to the last term (multiply)2Search and Seizure (Quad in AM)3Arrested and Caught (divide by ldquoardquo)4Beat Down (reduce fractions)5Parole (kick denominator to the front)6Check it out (FACE it)
Example
2xsup2 + 7x + 3
1xsup2 + 7x + 62(x + 6) (x + 1)3(x + 62) (x + 12)4(x + 3) (x +12)5(x + 3) (2x + 1)
No perfect squares and 3 terms and starts with xsup2
Quadratics in the Morning (AM)
1 Make a factor tree2 Multiply to last number add
to middle number3 Put out baggies (parenthesis)4 Split first term nice5 Drop in factors from tree
Example
xsup2 + 12x + 32
(x + 8) (x + 4)
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Solve with Graphing Calculator
1) Solve each equation for y (3 easy steps)
2) Use y= button and enter each equation
3) Use graph to eyeball answerOr
4) Use to find where y1 and y2 are equal
5) Be sure answer is in (x y) form
Y=
Systems of EquationsTo Solve by Graphing
1) Make an x y Chart
2) Select any x solve for y x 2x ndash 4 y 0 2(0) ndash 4 -4 1 2(1) ndash 4 -2
3) Then graph the two points4) Do for both equations5) The answer is where they
cross6) Be sure answer is in (x y)
form
2nd TABLE
bull y + 2x = 9
bull y = 9 ndash 2xbull 3(9 - 2x) ndash 2x = 11
bull 27-6x ndash 2x = 11bull 27 ndash 8x = 11bull - 8x = -16bull x = 2bull y = 9 ndash 2 (2)bull y = 5bull (2 5)
Systems of EquationsSolve by Substitution
(box amp shove)
1) Solve one equation for x or y (change sides change signs)
2) Box it3) Rewrite other equation
and shove box in 4) Solve for surviving letter
1) Distribute2) Combine like terms3) Solve
4) Send it back to box5) Solve for other letter6) Answer in (xy) form
3y -2x = 11y + 2x = 9
Systems of EquationsSolve by Elimination
1) Look for opposite signs
2) Multiply to create opposites
3) Add old and new equation together
4) Solve for surviving letter
5) Plug back into either equation (pick the easy one)
6) Solve for other letter
7) Answer in (xy) format
bull 2x ndash y = 93x + 4y = -14
bull 4(2x ndash y = 9)
bull 8x ndash 4y = 36bull 3x + 4y = -14bull 11x = 22bull x = 2
bull 2(2) ndash y = 9bull -y = 5bull y = -5
bull (2 -5)
2x ndash y = 93x + 4y = -14
Word Clue Word Sentence Algebraic
divide
Quotient
divided by
The quotient of a number and 710 divided by a number
xdivide7
10dividex
The first number written before the clue word will be the numerator
Clue Words for writing equations from word problems
Division
1 Consistent ndash one or many solutions2 Inconsistent ndash No solution
1 Independent ndash Only one solution2 Dependent ndash Has infinitely many
solutions
Slope Y ndash Int Graph Type of Systems
of Solutions
Same Same Same line ConsistentDependent
Infinitely many
Same Different Parallel Inconsistent 0
Different DifferentSame
Intersects ConsistentIndependent
1
Consistent and Inconsistent Systems
Slope ndash Intercept Form
y = mx + b
Slope- directions
RiseRun
Y Intercept ndash where to start
Itrsquos a line address
If the slope is a whole number put it on a stick m = 2 slope is 21
To Graph
y = 2X + 1 Starts at 1
Riserun = 21
Directions are up 2 over 1
Example 1 Example 2y = -3X+ 0
y = -3X
Starts at 0
riserun = 3-1
Directions are up 3 over -1
Thanks to httpwwwmathsisfuncomequation_of_linehtml
Linear Equations Standard Form ax + by = cSolving for y Just 3 easy stepsGreenisms Math Terms
1Move x term(Change sides Change signs) AddSubtract x term 2Give x side a hug Parenthesis ( )3Divide by number next to the y Coefficient
Example Solve for Y2x ndash 7y = 12Just 3 easy steps1 -7y = 12 ndash 2x (Move x term (Change sides Change signs)2 -7y = (12-2x) (Give it a hug)3 y = (12-2x) -7 (Divide by number next to the y)
Now you are ready to enter it into the calculator and graph it
WATCH YOUR SIGNS
Example Solve for Y2x ndash 7y = 12
Just 3 easy steps1 -7y = 12 ndash 2x X is offside Penalty change signs2 -7y = (12-2x) Huddle up ( )3 y = (12-2x) -7 Man on man defense
WATCH YOUR SIGNS
Linear Equations Standard Form ax + by = cSolving for y Itrsquos a Football Game
Y VS Everybody Else
Follow football rules
Play FootballY vs everybody else
Now you are ready to enter it into the calculator and graph it
l lines
Same Slope
Slopes are Negative Reciprocal
(Flip amp Change Sign)
PARALLEL LINES
y2 ndash y1
x2 ndash x1
or
y = mx + b
slope
Find Equation of the Line y = mx + b
I need slope (m) amp the y-intercept
(b)
MY ANSWER
y = x +
To find m ndash Solve the equation for y and use mor use the y2 ndash y1
x2 ndash x1 formula
To find b - Plug x y and m into the line equation and solve for b
(Just follow the Rules)
Base xsup2Exponent
Like Bases Exponent Example
Multiply Add
Power up Multiply
Divide Subtract s
Add No change 3asup2 + 5asup2 = 8asup2 asup2 + = asup2 +
Subtract No change 10asup2 - 4asup2 = 6asup2
5a
4
15 9 3a g y
6 4 5
3 7 4a x ka x k
5a
5a
5a
Exponents
=
(xsup2asup3g)(xasup2gsup3) = (xsup3 g )
( gsup3 y) sup3 =
asup3k xsup3
Negative ExponentsJust switch places and make exponent positive
5x5
1x
Example Switch Simplify
5 4 3
2 2 4x g ka x k
4 2
2 5 4 3g x
a x k k
4
2 3 7g
a x k
=
Xsup2
X X∙ = Xsup2 X
X times X is X squared
X
QuadraticsMultiplying Binomials ndash Draw the Face
(x + 8) (x ndash 6)
Multiply Watch your Signs
xsup2 +8x -6x -48
Simplify (Combine Like Terms eat the buggers)xsup2 + 2x -48
Check This First
The Math ldquoFrdquo Word ldquoFactoringrdquo
1 Is there a common factor (number or letter)
NO
Proceed to question 2
Yes
Proceed with CGF
4xsup3 2 2 x x x
4x ( 2xy + xsup2 -3 )
Example
8xsup2y + 4xsup3 - 12x
Factor each term8xsup2y 2 2 2 x x y 12x 2 2 3 x
Circle common terms
8xsup2y 2 2 2 x x y 4xsup3 2 2 x x x 12x 2 2 3 x
Multiply circled numbersThatrsquos your Common Factor
Multiply leftovers put in ( )
2 Perfect squares on end amp 3 terms
xsup2 - 10x + 25
YES SPLIT IT NICE
NO proceed to question 3
xsup2 - 10x + 25Put out baggies to hold the answer (parenthesis)
( x - 5 ) ( x - 5 )
Split the first term nice
Place in baggies
Sign between is same as middle term
Split the second term nice
Place in baggies
ANSWER
(x -5)sup2
3 Perfect squares on end amp 2 terms
YesSame as question 2Except the signsare +-
Example
xsup2 - 64
NO proceed to question 4
( x - 8 ) ( x + 8)
Answer looks like this
4 No perfect squares on end 3 terms amp starts with xsup2
NO proceed to question 5
YES its quadratics in the morning
xsup2 - 10x + 24MA
Multiply to +24 Add to -10
38 11
-3 -8 -11
-2 -12 -14
-6-4 -10 Found it
Put in the parenthesis
(x-6) (x-4)
All Done
2 Search and Seizure (quadratics in the morning)(See question 4)
1 Steal the ldquoardquo and give it to the last term (Multiply)1
5 Is there a number in front of the xsup2 amp does it have 3 terms
Yes Jail BreakNO
Then it is Prime
(canrsquot factor)
Example 2xsup2 + 7x + 3
xsup2 + 7x + 6 (23)
( x + 6 ) ( x + 1 )
3 Arrested and Caught - Divide last terms by ldquoardquo
( x + 62 ) ( x + 12 )
4 Beat it Down - (reduce fractions)
( x + 3 ) ( x + 12 )
5 Parole (kick denominator to the front)(x + 3 ) ( 2 x + 1 )
All Done
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Check This First
Is there a common factor (number or letter)
Greatest Common Factor (GCF)
1 Factor each term2 Circle common terms3 Multiply common terms4 Write it down5 Put out baggie for leftover6 Multiply leftovers for each
term7 Put in baggie ( )
Example
8xsup2y + 4xsup3 - 12x
4x ( 2xy + xsup2 - 3 )
Is there a Square on each End
Perfect Squares
1 Put out Parenthesis2 Split FIRST and LAST numbers nice3 Put in Signs
3 Different Kinds
A xsup2 - 16x + 64
(xndash 8) (x ndash 8)
B xsup2 + 18x + 81
(x + 9) (x + 9)
C xsup2 - 36
(x + 6) (x ndash 6)
Is there a number in front of the xsup2 and does it have 3 terms
Jail Break
1Steal the ldquoardquo and give it to the last term (multiply)2Search and Seizure (Quad in AM)3Arrested and Caught (divide by ldquoardquo)4Beat Down (reduce fractions)5Parole (kick denominator to the front)6Check it out (FACE it)
Example
2xsup2 + 7x + 3
1xsup2 + 7x + 62(x + 6) (x + 1)3(x + 62) (x + 12)4(x + 3) (x +12)5(x + 3) (2x + 1)
No perfect squares and 3 terms and starts with xsup2
Quadratics in the Morning (AM)
1 Make a factor tree2 Multiply to last number add
to middle number3 Put out baggies (parenthesis)4 Split first term nice5 Drop in factors from tree
Example
xsup2 + 12x + 32
(x + 8) (x + 4)
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Solve with Graphing Calculator
1) Solve each equation for y (3 easy steps)
2) Use y= button and enter each equation
3) Use graph to eyeball answerOr
4) Use to find where y1 and y2 are equal
5) Be sure answer is in (x y) form
Y=
Systems of EquationsTo Solve by Graphing
1) Make an x y Chart
2) Select any x solve for y x 2x ndash 4 y 0 2(0) ndash 4 -4 1 2(1) ndash 4 -2
3) Then graph the two points4) Do for both equations5) The answer is where they
cross6) Be sure answer is in (x y)
form
2nd TABLE
bull y + 2x = 9
bull y = 9 ndash 2xbull 3(9 - 2x) ndash 2x = 11
bull 27-6x ndash 2x = 11bull 27 ndash 8x = 11bull - 8x = -16bull x = 2bull y = 9 ndash 2 (2)bull y = 5bull (2 5)
Systems of EquationsSolve by Substitution
(box amp shove)
1) Solve one equation for x or y (change sides change signs)
2) Box it3) Rewrite other equation
and shove box in 4) Solve for surviving letter
1) Distribute2) Combine like terms3) Solve
4) Send it back to box5) Solve for other letter6) Answer in (xy) form
3y -2x = 11y + 2x = 9
Systems of EquationsSolve by Elimination
1) Look for opposite signs
2) Multiply to create opposites
3) Add old and new equation together
4) Solve for surviving letter
5) Plug back into either equation (pick the easy one)
6) Solve for other letter
7) Answer in (xy) format
bull 2x ndash y = 93x + 4y = -14
bull 4(2x ndash y = 9)
bull 8x ndash 4y = 36bull 3x + 4y = -14bull 11x = 22bull x = 2
bull 2(2) ndash y = 9bull -y = 5bull y = -5
bull (2 -5)
2x ndash y = 93x + 4y = -14
1 Consistent ndash one or many solutions2 Inconsistent ndash No solution
1 Independent ndash Only one solution2 Dependent ndash Has infinitely many
solutions
Slope Y ndash Int Graph Type of Systems
of Solutions
Same Same Same line ConsistentDependent
Infinitely many
Same Different Parallel Inconsistent 0
Different DifferentSame
Intersects ConsistentIndependent
1
Consistent and Inconsistent Systems
Slope ndash Intercept Form
y = mx + b
Slope- directions
RiseRun
Y Intercept ndash where to start
Itrsquos a line address
If the slope is a whole number put it on a stick m = 2 slope is 21
To Graph
y = 2X + 1 Starts at 1
Riserun = 21
Directions are up 2 over 1
Example 1 Example 2y = -3X+ 0
y = -3X
Starts at 0
riserun = 3-1
Directions are up 3 over -1
Thanks to httpwwwmathsisfuncomequation_of_linehtml
Linear Equations Standard Form ax + by = cSolving for y Just 3 easy stepsGreenisms Math Terms
1Move x term(Change sides Change signs) AddSubtract x term 2Give x side a hug Parenthesis ( )3Divide by number next to the y Coefficient
Example Solve for Y2x ndash 7y = 12Just 3 easy steps1 -7y = 12 ndash 2x (Move x term (Change sides Change signs)2 -7y = (12-2x) (Give it a hug)3 y = (12-2x) -7 (Divide by number next to the y)
Now you are ready to enter it into the calculator and graph it
WATCH YOUR SIGNS
Example Solve for Y2x ndash 7y = 12
Just 3 easy steps1 -7y = 12 ndash 2x X is offside Penalty change signs2 -7y = (12-2x) Huddle up ( )3 y = (12-2x) -7 Man on man defense
WATCH YOUR SIGNS
Linear Equations Standard Form ax + by = cSolving for y Itrsquos a Football Game
Y VS Everybody Else
Follow football rules
Play FootballY vs everybody else
Now you are ready to enter it into the calculator and graph it
l lines
Same Slope
Slopes are Negative Reciprocal
(Flip amp Change Sign)
PARALLEL LINES
y2 ndash y1
x2 ndash x1
or
y = mx + b
slope
Find Equation of the Line y = mx + b
I need slope (m) amp the y-intercept
(b)
MY ANSWER
y = x +
To find m ndash Solve the equation for y and use mor use the y2 ndash y1
x2 ndash x1 formula
To find b - Plug x y and m into the line equation and solve for b
(Just follow the Rules)
Base xsup2Exponent
Like Bases Exponent Example
Multiply Add
Power up Multiply
Divide Subtract s
Add No change 3asup2 + 5asup2 = 8asup2 asup2 + = asup2 +
Subtract No change 10asup2 - 4asup2 = 6asup2
5a
4
15 9 3a g y
6 4 5
3 7 4a x ka x k
5a
5a
5a
Exponents
=
(xsup2asup3g)(xasup2gsup3) = (xsup3 g )
( gsup3 y) sup3 =
asup3k xsup3
Negative ExponentsJust switch places and make exponent positive
5x5
1x
Example Switch Simplify
5 4 3
2 2 4x g ka x k
4 2
2 5 4 3g x
a x k k
4
2 3 7g
a x k
=
Xsup2
X X∙ = Xsup2 X
X times X is X squared
X
QuadraticsMultiplying Binomials ndash Draw the Face
(x + 8) (x ndash 6)
Multiply Watch your Signs
xsup2 +8x -6x -48
Simplify (Combine Like Terms eat the buggers)xsup2 + 2x -48
Check This First
The Math ldquoFrdquo Word ldquoFactoringrdquo
1 Is there a common factor (number or letter)
NO
Proceed to question 2
Yes
Proceed with CGF
4xsup3 2 2 x x x
4x ( 2xy + xsup2 -3 )
Example
8xsup2y + 4xsup3 - 12x
Factor each term8xsup2y 2 2 2 x x y 12x 2 2 3 x
Circle common terms
8xsup2y 2 2 2 x x y 4xsup3 2 2 x x x 12x 2 2 3 x
Multiply circled numbersThatrsquos your Common Factor
Multiply leftovers put in ( )
2 Perfect squares on end amp 3 terms
xsup2 - 10x + 25
YES SPLIT IT NICE
NO proceed to question 3
xsup2 - 10x + 25Put out baggies to hold the answer (parenthesis)
( x - 5 ) ( x - 5 )
Split the first term nice
Place in baggies
Sign between is same as middle term
Split the second term nice
Place in baggies
ANSWER
(x -5)sup2
3 Perfect squares on end amp 2 terms
YesSame as question 2Except the signsare +-
Example
xsup2 - 64
NO proceed to question 4
( x - 8 ) ( x + 8)
Answer looks like this
4 No perfect squares on end 3 terms amp starts with xsup2
NO proceed to question 5
YES its quadratics in the morning
xsup2 - 10x + 24MA
Multiply to +24 Add to -10
38 11
-3 -8 -11
-2 -12 -14
-6-4 -10 Found it
Put in the parenthesis
(x-6) (x-4)
All Done
2 Search and Seizure (quadratics in the morning)(See question 4)
1 Steal the ldquoardquo and give it to the last term (Multiply)1
5 Is there a number in front of the xsup2 amp does it have 3 terms
Yes Jail BreakNO
Then it is Prime
(canrsquot factor)
Example 2xsup2 + 7x + 3
xsup2 + 7x + 6 (23)
( x + 6 ) ( x + 1 )
3 Arrested and Caught - Divide last terms by ldquoardquo
( x + 62 ) ( x + 12 )
4 Beat it Down - (reduce fractions)
( x + 3 ) ( x + 12 )
5 Parole (kick denominator to the front)(x + 3 ) ( 2 x + 1 )
All Done
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Check This First
Is there a common factor (number or letter)
Greatest Common Factor (GCF)
1 Factor each term2 Circle common terms3 Multiply common terms4 Write it down5 Put out baggie for leftover6 Multiply leftovers for each
term7 Put in baggie ( )
Example
8xsup2y + 4xsup3 - 12x
4x ( 2xy + xsup2 - 3 )
Is there a Square on each End
Perfect Squares
1 Put out Parenthesis2 Split FIRST and LAST numbers nice3 Put in Signs
3 Different Kinds
A xsup2 - 16x + 64
(xndash 8) (x ndash 8)
B xsup2 + 18x + 81
(x + 9) (x + 9)
C xsup2 - 36
(x + 6) (x ndash 6)
Is there a number in front of the xsup2 and does it have 3 terms
Jail Break
1Steal the ldquoardquo and give it to the last term (multiply)2Search and Seizure (Quad in AM)3Arrested and Caught (divide by ldquoardquo)4Beat Down (reduce fractions)5Parole (kick denominator to the front)6Check it out (FACE it)
Example
2xsup2 + 7x + 3
1xsup2 + 7x + 62(x + 6) (x + 1)3(x + 62) (x + 12)4(x + 3) (x +12)5(x + 3) (2x + 1)
No perfect squares and 3 terms and starts with xsup2
Quadratics in the Morning (AM)
1 Make a factor tree2 Multiply to last number add
to middle number3 Put out baggies (parenthesis)4 Split first term nice5 Drop in factors from tree
Example
xsup2 + 12x + 32
(x + 8) (x + 4)
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Solve with Graphing Calculator
1) Solve each equation for y (3 easy steps)
2) Use y= button and enter each equation
3) Use graph to eyeball answerOr
4) Use to find where y1 and y2 are equal
5) Be sure answer is in (x y) form
Y=
Systems of EquationsTo Solve by Graphing
1) Make an x y Chart
2) Select any x solve for y x 2x ndash 4 y 0 2(0) ndash 4 -4 1 2(1) ndash 4 -2
3) Then graph the two points4) Do for both equations5) The answer is where they
cross6) Be sure answer is in (x y)
form
2nd TABLE
bull y + 2x = 9
bull y = 9 ndash 2xbull 3(9 - 2x) ndash 2x = 11
bull 27-6x ndash 2x = 11bull 27 ndash 8x = 11bull - 8x = -16bull x = 2bull y = 9 ndash 2 (2)bull y = 5bull (2 5)
Systems of EquationsSolve by Substitution
(box amp shove)
1) Solve one equation for x or y (change sides change signs)
2) Box it3) Rewrite other equation
and shove box in 4) Solve for surviving letter
1) Distribute2) Combine like terms3) Solve
4) Send it back to box5) Solve for other letter6) Answer in (xy) form
3y -2x = 11y + 2x = 9
Systems of EquationsSolve by Elimination
1) Look for opposite signs
2) Multiply to create opposites
3) Add old and new equation together
4) Solve for surviving letter
5) Plug back into either equation (pick the easy one)
6) Solve for other letter
7) Answer in (xy) format
bull 2x ndash y = 93x + 4y = -14
bull 4(2x ndash y = 9)
bull 8x ndash 4y = 36bull 3x + 4y = -14bull 11x = 22bull x = 2
bull 2(2) ndash y = 9bull -y = 5bull y = -5
bull (2 -5)
2x ndash y = 93x + 4y = -14
Slope ndash Intercept Form
y = mx + b
Slope- directions
RiseRun
Y Intercept ndash where to start
Itrsquos a line address
If the slope is a whole number put it on a stick m = 2 slope is 21
To Graph
y = 2X + 1 Starts at 1
Riserun = 21
Directions are up 2 over 1
Example 1 Example 2y = -3X+ 0
y = -3X
Starts at 0
riserun = 3-1
Directions are up 3 over -1
Thanks to httpwwwmathsisfuncomequation_of_linehtml
Linear Equations Standard Form ax + by = cSolving for y Just 3 easy stepsGreenisms Math Terms
1Move x term(Change sides Change signs) AddSubtract x term 2Give x side a hug Parenthesis ( )3Divide by number next to the y Coefficient
Example Solve for Y2x ndash 7y = 12Just 3 easy steps1 -7y = 12 ndash 2x (Move x term (Change sides Change signs)2 -7y = (12-2x) (Give it a hug)3 y = (12-2x) -7 (Divide by number next to the y)
Now you are ready to enter it into the calculator and graph it
WATCH YOUR SIGNS
Example Solve for Y2x ndash 7y = 12
Just 3 easy steps1 -7y = 12 ndash 2x X is offside Penalty change signs2 -7y = (12-2x) Huddle up ( )3 y = (12-2x) -7 Man on man defense
WATCH YOUR SIGNS
Linear Equations Standard Form ax + by = cSolving for y Itrsquos a Football Game
Y VS Everybody Else
Follow football rules
Play FootballY vs everybody else
Now you are ready to enter it into the calculator and graph it
l lines
Same Slope
Slopes are Negative Reciprocal
(Flip amp Change Sign)
PARALLEL LINES
y2 ndash y1
x2 ndash x1
or
y = mx + b
slope
Find Equation of the Line y = mx + b
I need slope (m) amp the y-intercept
(b)
MY ANSWER
y = x +
To find m ndash Solve the equation for y and use mor use the y2 ndash y1
x2 ndash x1 formula
To find b - Plug x y and m into the line equation and solve for b
(Just follow the Rules)
Base xsup2Exponent
Like Bases Exponent Example
Multiply Add
Power up Multiply
Divide Subtract s
Add No change 3asup2 + 5asup2 = 8asup2 asup2 + = asup2 +
Subtract No change 10asup2 - 4asup2 = 6asup2
5a
4
15 9 3a g y
6 4 5
3 7 4a x ka x k
5a
5a
5a
Exponents
=
(xsup2asup3g)(xasup2gsup3) = (xsup3 g )
( gsup3 y) sup3 =
asup3k xsup3
Negative ExponentsJust switch places and make exponent positive
5x5
1x
Example Switch Simplify
5 4 3
2 2 4x g ka x k
4 2
2 5 4 3g x
a x k k
4
2 3 7g
a x k
=
Xsup2
X X∙ = Xsup2 X
X times X is X squared
X
QuadraticsMultiplying Binomials ndash Draw the Face
(x + 8) (x ndash 6)
Multiply Watch your Signs
xsup2 +8x -6x -48
Simplify (Combine Like Terms eat the buggers)xsup2 + 2x -48
Check This First
The Math ldquoFrdquo Word ldquoFactoringrdquo
1 Is there a common factor (number or letter)
NO
Proceed to question 2
Yes
Proceed with CGF
4xsup3 2 2 x x x
4x ( 2xy + xsup2 -3 )
Example
8xsup2y + 4xsup3 - 12x
Factor each term8xsup2y 2 2 2 x x y 12x 2 2 3 x
Circle common terms
8xsup2y 2 2 2 x x y 4xsup3 2 2 x x x 12x 2 2 3 x
Multiply circled numbersThatrsquos your Common Factor
Multiply leftovers put in ( )
2 Perfect squares on end amp 3 terms
xsup2 - 10x + 25
YES SPLIT IT NICE
NO proceed to question 3
xsup2 - 10x + 25Put out baggies to hold the answer (parenthesis)
( x - 5 ) ( x - 5 )
Split the first term nice
Place in baggies
Sign between is same as middle term
Split the second term nice
Place in baggies
ANSWER
(x -5)sup2
3 Perfect squares on end amp 2 terms
YesSame as question 2Except the signsare +-
Example
xsup2 - 64
NO proceed to question 4
( x - 8 ) ( x + 8)
Answer looks like this
4 No perfect squares on end 3 terms amp starts with xsup2
NO proceed to question 5
YES its quadratics in the morning
xsup2 - 10x + 24MA
Multiply to +24 Add to -10
38 11
-3 -8 -11
-2 -12 -14
-6-4 -10 Found it
Put in the parenthesis
(x-6) (x-4)
All Done
2 Search and Seizure (quadratics in the morning)(See question 4)
1 Steal the ldquoardquo and give it to the last term (Multiply)1
5 Is there a number in front of the xsup2 amp does it have 3 terms
Yes Jail BreakNO
Then it is Prime
(canrsquot factor)
Example 2xsup2 + 7x + 3
xsup2 + 7x + 6 (23)
( x + 6 ) ( x + 1 )
3 Arrested and Caught - Divide last terms by ldquoardquo
( x + 62 ) ( x + 12 )
4 Beat it Down - (reduce fractions)
( x + 3 ) ( x + 12 )
5 Parole (kick denominator to the front)(x + 3 ) ( 2 x + 1 )
All Done
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Check This First
Is there a common factor (number or letter)
Greatest Common Factor (GCF)
1 Factor each term2 Circle common terms3 Multiply common terms4 Write it down5 Put out baggie for leftover6 Multiply leftovers for each
term7 Put in baggie ( )
Example
8xsup2y + 4xsup3 - 12x
4x ( 2xy + xsup2 - 3 )
Is there a Square on each End
Perfect Squares
1 Put out Parenthesis2 Split FIRST and LAST numbers nice3 Put in Signs
3 Different Kinds
A xsup2 - 16x + 64
(xndash 8) (x ndash 8)
B xsup2 + 18x + 81
(x + 9) (x + 9)
C xsup2 - 36
(x + 6) (x ndash 6)
Is there a number in front of the xsup2 and does it have 3 terms
Jail Break
1Steal the ldquoardquo and give it to the last term (multiply)2Search and Seizure (Quad in AM)3Arrested and Caught (divide by ldquoardquo)4Beat Down (reduce fractions)5Parole (kick denominator to the front)6Check it out (FACE it)
Example
2xsup2 + 7x + 3
1xsup2 + 7x + 62(x + 6) (x + 1)3(x + 62) (x + 12)4(x + 3) (x +12)5(x + 3) (2x + 1)
No perfect squares and 3 terms and starts with xsup2
Quadratics in the Morning (AM)
1 Make a factor tree2 Multiply to last number add
to middle number3 Put out baggies (parenthesis)4 Split first term nice5 Drop in factors from tree
Example
xsup2 + 12x + 32
(x + 8) (x + 4)
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Solve with Graphing Calculator
1) Solve each equation for y (3 easy steps)
2) Use y= button and enter each equation
3) Use graph to eyeball answerOr
4) Use to find where y1 and y2 are equal
5) Be sure answer is in (x y) form
Y=
Systems of EquationsTo Solve by Graphing
1) Make an x y Chart
2) Select any x solve for y x 2x ndash 4 y 0 2(0) ndash 4 -4 1 2(1) ndash 4 -2
3) Then graph the two points4) Do for both equations5) The answer is where they
cross6) Be sure answer is in (x y)
form
2nd TABLE
bull y + 2x = 9
bull y = 9 ndash 2xbull 3(9 - 2x) ndash 2x = 11
bull 27-6x ndash 2x = 11bull 27 ndash 8x = 11bull - 8x = -16bull x = 2bull y = 9 ndash 2 (2)bull y = 5bull (2 5)
Systems of EquationsSolve by Substitution
(box amp shove)
1) Solve one equation for x or y (change sides change signs)
2) Box it3) Rewrite other equation
and shove box in 4) Solve for surviving letter
1) Distribute2) Combine like terms3) Solve
4) Send it back to box5) Solve for other letter6) Answer in (xy) form
3y -2x = 11y + 2x = 9
Systems of EquationsSolve by Elimination
1) Look for opposite signs
2) Multiply to create opposites
3) Add old and new equation together
4) Solve for surviving letter
5) Plug back into either equation (pick the easy one)
6) Solve for other letter
7) Answer in (xy) format
bull 2x ndash y = 93x + 4y = -14
bull 4(2x ndash y = 9)
bull 8x ndash 4y = 36bull 3x + 4y = -14bull 11x = 22bull x = 2
bull 2(2) ndash y = 9bull -y = 5bull y = -5
bull (2 -5)
2x ndash y = 93x + 4y = -14
Linear Equations Standard Form ax + by = cSolving for y Just 3 easy stepsGreenisms Math Terms
1Move x term(Change sides Change signs) AddSubtract x term 2Give x side a hug Parenthesis ( )3Divide by number next to the y Coefficient
Example Solve for Y2x ndash 7y = 12Just 3 easy steps1 -7y = 12 ndash 2x (Move x term (Change sides Change signs)2 -7y = (12-2x) (Give it a hug)3 y = (12-2x) -7 (Divide by number next to the y)
Now you are ready to enter it into the calculator and graph it
WATCH YOUR SIGNS
Example Solve for Y2x ndash 7y = 12
Just 3 easy steps1 -7y = 12 ndash 2x X is offside Penalty change signs2 -7y = (12-2x) Huddle up ( )3 y = (12-2x) -7 Man on man defense
WATCH YOUR SIGNS
Linear Equations Standard Form ax + by = cSolving for y Itrsquos a Football Game
Y VS Everybody Else
Follow football rules
Play FootballY vs everybody else
Now you are ready to enter it into the calculator and graph it
l lines
Same Slope
Slopes are Negative Reciprocal
(Flip amp Change Sign)
PARALLEL LINES
y2 ndash y1
x2 ndash x1
or
y = mx + b
slope
Find Equation of the Line y = mx + b
I need slope (m) amp the y-intercept
(b)
MY ANSWER
y = x +
To find m ndash Solve the equation for y and use mor use the y2 ndash y1
x2 ndash x1 formula
To find b - Plug x y and m into the line equation and solve for b
(Just follow the Rules)
Base xsup2Exponent
Like Bases Exponent Example
Multiply Add
Power up Multiply
Divide Subtract s
Add No change 3asup2 + 5asup2 = 8asup2 asup2 + = asup2 +
Subtract No change 10asup2 - 4asup2 = 6asup2
5a
4
15 9 3a g y
6 4 5
3 7 4a x ka x k
5a
5a
5a
Exponents
=
(xsup2asup3g)(xasup2gsup3) = (xsup3 g )
( gsup3 y) sup3 =
asup3k xsup3
Negative ExponentsJust switch places and make exponent positive
5x5
1x
Example Switch Simplify
5 4 3
2 2 4x g ka x k
4 2
2 5 4 3g x
a x k k
4
2 3 7g
a x k
=
Xsup2
X X∙ = Xsup2 X
X times X is X squared
X
QuadraticsMultiplying Binomials ndash Draw the Face
(x + 8) (x ndash 6)
Multiply Watch your Signs
xsup2 +8x -6x -48
Simplify (Combine Like Terms eat the buggers)xsup2 + 2x -48
Check This First
The Math ldquoFrdquo Word ldquoFactoringrdquo
1 Is there a common factor (number or letter)
NO
Proceed to question 2
Yes
Proceed with CGF
4xsup3 2 2 x x x
4x ( 2xy + xsup2 -3 )
Example
8xsup2y + 4xsup3 - 12x
Factor each term8xsup2y 2 2 2 x x y 12x 2 2 3 x
Circle common terms
8xsup2y 2 2 2 x x y 4xsup3 2 2 x x x 12x 2 2 3 x
Multiply circled numbersThatrsquos your Common Factor
Multiply leftovers put in ( )
2 Perfect squares on end amp 3 terms
xsup2 - 10x + 25
YES SPLIT IT NICE
NO proceed to question 3
xsup2 - 10x + 25Put out baggies to hold the answer (parenthesis)
( x - 5 ) ( x - 5 )
Split the first term nice
Place in baggies
Sign between is same as middle term
Split the second term nice
Place in baggies
ANSWER
(x -5)sup2
3 Perfect squares on end amp 2 terms
YesSame as question 2Except the signsare +-
Example
xsup2 - 64
NO proceed to question 4
( x - 8 ) ( x + 8)
Answer looks like this
4 No perfect squares on end 3 terms amp starts with xsup2
NO proceed to question 5
YES its quadratics in the morning
xsup2 - 10x + 24MA
Multiply to +24 Add to -10
38 11
-3 -8 -11
-2 -12 -14
-6-4 -10 Found it
Put in the parenthesis
(x-6) (x-4)
All Done
2 Search and Seizure (quadratics in the morning)(See question 4)
1 Steal the ldquoardquo and give it to the last term (Multiply)1
5 Is there a number in front of the xsup2 amp does it have 3 terms
Yes Jail BreakNO
Then it is Prime
(canrsquot factor)
Example 2xsup2 + 7x + 3
xsup2 + 7x + 6 (23)
( x + 6 ) ( x + 1 )
3 Arrested and Caught - Divide last terms by ldquoardquo
( x + 62 ) ( x + 12 )
4 Beat it Down - (reduce fractions)
( x + 3 ) ( x + 12 )
5 Parole (kick denominator to the front)(x + 3 ) ( 2 x + 1 )
All Done
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Check This First
Is there a common factor (number or letter)
Greatest Common Factor (GCF)
1 Factor each term2 Circle common terms3 Multiply common terms4 Write it down5 Put out baggie for leftover6 Multiply leftovers for each
term7 Put in baggie ( )
Example
8xsup2y + 4xsup3 - 12x
4x ( 2xy + xsup2 - 3 )
Is there a Square on each End
Perfect Squares
1 Put out Parenthesis2 Split FIRST and LAST numbers nice3 Put in Signs
3 Different Kinds
A xsup2 - 16x + 64
(xndash 8) (x ndash 8)
B xsup2 + 18x + 81
(x + 9) (x + 9)
C xsup2 - 36
(x + 6) (x ndash 6)
Is there a number in front of the xsup2 and does it have 3 terms
Jail Break
1Steal the ldquoardquo and give it to the last term (multiply)2Search and Seizure (Quad in AM)3Arrested and Caught (divide by ldquoardquo)4Beat Down (reduce fractions)5Parole (kick denominator to the front)6Check it out (FACE it)
Example
2xsup2 + 7x + 3
1xsup2 + 7x + 62(x + 6) (x + 1)3(x + 62) (x + 12)4(x + 3) (x +12)5(x + 3) (2x + 1)
No perfect squares and 3 terms and starts with xsup2
Quadratics in the Morning (AM)
1 Make a factor tree2 Multiply to last number add
to middle number3 Put out baggies (parenthesis)4 Split first term nice5 Drop in factors from tree
Example
xsup2 + 12x + 32
(x + 8) (x + 4)
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Solve with Graphing Calculator
1) Solve each equation for y (3 easy steps)
2) Use y= button and enter each equation
3) Use graph to eyeball answerOr
4) Use to find where y1 and y2 are equal
5) Be sure answer is in (x y) form
Y=
Systems of EquationsTo Solve by Graphing
1) Make an x y Chart
2) Select any x solve for y x 2x ndash 4 y 0 2(0) ndash 4 -4 1 2(1) ndash 4 -2
3) Then graph the two points4) Do for both equations5) The answer is where they
cross6) Be sure answer is in (x y)
form
2nd TABLE
bull y + 2x = 9
bull y = 9 ndash 2xbull 3(9 - 2x) ndash 2x = 11
bull 27-6x ndash 2x = 11bull 27 ndash 8x = 11bull - 8x = -16bull x = 2bull y = 9 ndash 2 (2)bull y = 5bull (2 5)
Systems of EquationsSolve by Substitution
(box amp shove)
1) Solve one equation for x or y (change sides change signs)
2) Box it3) Rewrite other equation
and shove box in 4) Solve for surviving letter
1) Distribute2) Combine like terms3) Solve
4) Send it back to box5) Solve for other letter6) Answer in (xy) form
3y -2x = 11y + 2x = 9
Systems of EquationsSolve by Elimination
1) Look for opposite signs
2) Multiply to create opposites
3) Add old and new equation together
4) Solve for surviving letter
5) Plug back into either equation (pick the easy one)
6) Solve for other letter
7) Answer in (xy) format
bull 2x ndash y = 93x + 4y = -14
bull 4(2x ndash y = 9)
bull 8x ndash 4y = 36bull 3x + 4y = -14bull 11x = 22bull x = 2
bull 2(2) ndash y = 9bull -y = 5bull y = -5
bull (2 -5)
2x ndash y = 93x + 4y = -14
Example Solve for Y2x ndash 7y = 12
Just 3 easy steps1 -7y = 12 ndash 2x X is offside Penalty change signs2 -7y = (12-2x) Huddle up ( )3 y = (12-2x) -7 Man on man defense
WATCH YOUR SIGNS
Linear Equations Standard Form ax + by = cSolving for y Itrsquos a Football Game
Y VS Everybody Else
Follow football rules
Play FootballY vs everybody else
Now you are ready to enter it into the calculator and graph it
l lines
Same Slope
Slopes are Negative Reciprocal
(Flip amp Change Sign)
PARALLEL LINES
y2 ndash y1
x2 ndash x1
or
y = mx + b
slope
Find Equation of the Line y = mx + b
I need slope (m) amp the y-intercept
(b)
MY ANSWER
y = x +
To find m ndash Solve the equation for y and use mor use the y2 ndash y1
x2 ndash x1 formula
To find b - Plug x y and m into the line equation and solve for b
(Just follow the Rules)
Base xsup2Exponent
Like Bases Exponent Example
Multiply Add
Power up Multiply
Divide Subtract s
Add No change 3asup2 + 5asup2 = 8asup2 asup2 + = asup2 +
Subtract No change 10asup2 - 4asup2 = 6asup2
5a
4
15 9 3a g y
6 4 5
3 7 4a x ka x k
5a
5a
5a
Exponents
=
(xsup2asup3g)(xasup2gsup3) = (xsup3 g )
( gsup3 y) sup3 =
asup3k xsup3
Negative ExponentsJust switch places and make exponent positive
5x5
1x
Example Switch Simplify
5 4 3
2 2 4x g ka x k
4 2
2 5 4 3g x
a x k k
4
2 3 7g
a x k
=
Xsup2
X X∙ = Xsup2 X
X times X is X squared
X
QuadraticsMultiplying Binomials ndash Draw the Face
(x + 8) (x ndash 6)
Multiply Watch your Signs
xsup2 +8x -6x -48
Simplify (Combine Like Terms eat the buggers)xsup2 + 2x -48
Check This First
The Math ldquoFrdquo Word ldquoFactoringrdquo
1 Is there a common factor (number or letter)
NO
Proceed to question 2
Yes
Proceed with CGF
4xsup3 2 2 x x x
4x ( 2xy + xsup2 -3 )
Example
8xsup2y + 4xsup3 - 12x
Factor each term8xsup2y 2 2 2 x x y 12x 2 2 3 x
Circle common terms
8xsup2y 2 2 2 x x y 4xsup3 2 2 x x x 12x 2 2 3 x
Multiply circled numbersThatrsquos your Common Factor
Multiply leftovers put in ( )
2 Perfect squares on end amp 3 terms
xsup2 - 10x + 25
YES SPLIT IT NICE
NO proceed to question 3
xsup2 - 10x + 25Put out baggies to hold the answer (parenthesis)
( x - 5 ) ( x - 5 )
Split the first term nice
Place in baggies
Sign between is same as middle term
Split the second term nice
Place in baggies
ANSWER
(x -5)sup2
3 Perfect squares on end amp 2 terms
YesSame as question 2Except the signsare +-
Example
xsup2 - 64
NO proceed to question 4
( x - 8 ) ( x + 8)
Answer looks like this
4 No perfect squares on end 3 terms amp starts with xsup2
NO proceed to question 5
YES its quadratics in the morning
xsup2 - 10x + 24MA
Multiply to +24 Add to -10
38 11
-3 -8 -11
-2 -12 -14
-6-4 -10 Found it
Put in the parenthesis
(x-6) (x-4)
All Done
2 Search and Seizure (quadratics in the morning)(See question 4)
1 Steal the ldquoardquo and give it to the last term (Multiply)1
5 Is there a number in front of the xsup2 amp does it have 3 terms
Yes Jail BreakNO
Then it is Prime
(canrsquot factor)
Example 2xsup2 + 7x + 3
xsup2 + 7x + 6 (23)
( x + 6 ) ( x + 1 )
3 Arrested and Caught - Divide last terms by ldquoardquo
( x + 62 ) ( x + 12 )
4 Beat it Down - (reduce fractions)
( x + 3 ) ( x + 12 )
5 Parole (kick denominator to the front)(x + 3 ) ( 2 x + 1 )
All Done
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Check This First
Is there a common factor (number or letter)
Greatest Common Factor (GCF)
1 Factor each term2 Circle common terms3 Multiply common terms4 Write it down5 Put out baggie for leftover6 Multiply leftovers for each
term7 Put in baggie ( )
Example
8xsup2y + 4xsup3 - 12x
4x ( 2xy + xsup2 - 3 )
Is there a Square on each End
Perfect Squares
1 Put out Parenthesis2 Split FIRST and LAST numbers nice3 Put in Signs
3 Different Kinds
A xsup2 - 16x + 64
(xndash 8) (x ndash 8)
B xsup2 + 18x + 81
(x + 9) (x + 9)
C xsup2 - 36
(x + 6) (x ndash 6)
Is there a number in front of the xsup2 and does it have 3 terms
Jail Break
1Steal the ldquoardquo and give it to the last term (multiply)2Search and Seizure (Quad in AM)3Arrested and Caught (divide by ldquoardquo)4Beat Down (reduce fractions)5Parole (kick denominator to the front)6Check it out (FACE it)
Example
2xsup2 + 7x + 3
1xsup2 + 7x + 62(x + 6) (x + 1)3(x + 62) (x + 12)4(x + 3) (x +12)5(x + 3) (2x + 1)
No perfect squares and 3 terms and starts with xsup2
Quadratics in the Morning (AM)
1 Make a factor tree2 Multiply to last number add
to middle number3 Put out baggies (parenthesis)4 Split first term nice5 Drop in factors from tree
Example
xsup2 + 12x + 32
(x + 8) (x + 4)
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Solve with Graphing Calculator
1) Solve each equation for y (3 easy steps)
2) Use y= button and enter each equation
3) Use graph to eyeball answerOr
4) Use to find where y1 and y2 are equal
5) Be sure answer is in (x y) form
Y=
Systems of EquationsTo Solve by Graphing
1) Make an x y Chart
2) Select any x solve for y x 2x ndash 4 y 0 2(0) ndash 4 -4 1 2(1) ndash 4 -2
3) Then graph the two points4) Do for both equations5) The answer is where they
cross6) Be sure answer is in (x y)
form
2nd TABLE
bull y + 2x = 9
bull y = 9 ndash 2xbull 3(9 - 2x) ndash 2x = 11
bull 27-6x ndash 2x = 11bull 27 ndash 8x = 11bull - 8x = -16bull x = 2bull y = 9 ndash 2 (2)bull y = 5bull (2 5)
Systems of EquationsSolve by Substitution
(box amp shove)
1) Solve one equation for x or y (change sides change signs)
2) Box it3) Rewrite other equation
and shove box in 4) Solve for surviving letter
1) Distribute2) Combine like terms3) Solve
4) Send it back to box5) Solve for other letter6) Answer in (xy) form
3y -2x = 11y + 2x = 9
Systems of EquationsSolve by Elimination
1) Look for opposite signs
2) Multiply to create opposites
3) Add old and new equation together
4) Solve for surviving letter
5) Plug back into either equation (pick the easy one)
6) Solve for other letter
7) Answer in (xy) format
bull 2x ndash y = 93x + 4y = -14
bull 4(2x ndash y = 9)
bull 8x ndash 4y = 36bull 3x + 4y = -14bull 11x = 22bull x = 2
bull 2(2) ndash y = 9bull -y = 5bull y = -5
bull (2 -5)
2x ndash y = 93x + 4y = -14
l lines
Same Slope
Slopes are Negative Reciprocal
(Flip amp Change Sign)
PARALLEL LINES
y2 ndash y1
x2 ndash x1
or
y = mx + b
slope
Find Equation of the Line y = mx + b
I need slope (m) amp the y-intercept
(b)
MY ANSWER
y = x +
To find m ndash Solve the equation for y and use mor use the y2 ndash y1
x2 ndash x1 formula
To find b - Plug x y and m into the line equation and solve for b
(Just follow the Rules)
Base xsup2Exponent
Like Bases Exponent Example
Multiply Add
Power up Multiply
Divide Subtract s
Add No change 3asup2 + 5asup2 = 8asup2 asup2 + = asup2 +
Subtract No change 10asup2 - 4asup2 = 6asup2
5a
4
15 9 3a g y
6 4 5
3 7 4a x ka x k
5a
5a
5a
Exponents
=
(xsup2asup3g)(xasup2gsup3) = (xsup3 g )
( gsup3 y) sup3 =
asup3k xsup3
Negative ExponentsJust switch places and make exponent positive
5x5
1x
Example Switch Simplify
5 4 3
2 2 4x g ka x k
4 2
2 5 4 3g x
a x k k
4
2 3 7g
a x k
=
Xsup2
X X∙ = Xsup2 X
X times X is X squared
X
QuadraticsMultiplying Binomials ndash Draw the Face
(x + 8) (x ndash 6)
Multiply Watch your Signs
xsup2 +8x -6x -48
Simplify (Combine Like Terms eat the buggers)xsup2 + 2x -48
Check This First
The Math ldquoFrdquo Word ldquoFactoringrdquo
1 Is there a common factor (number or letter)
NO
Proceed to question 2
Yes
Proceed with CGF
4xsup3 2 2 x x x
4x ( 2xy + xsup2 -3 )
Example
8xsup2y + 4xsup3 - 12x
Factor each term8xsup2y 2 2 2 x x y 12x 2 2 3 x
Circle common terms
8xsup2y 2 2 2 x x y 4xsup3 2 2 x x x 12x 2 2 3 x
Multiply circled numbersThatrsquos your Common Factor
Multiply leftovers put in ( )
2 Perfect squares on end amp 3 terms
xsup2 - 10x + 25
YES SPLIT IT NICE
NO proceed to question 3
xsup2 - 10x + 25Put out baggies to hold the answer (parenthesis)
( x - 5 ) ( x - 5 )
Split the first term nice
Place in baggies
Sign between is same as middle term
Split the second term nice
Place in baggies
ANSWER
(x -5)sup2
3 Perfect squares on end amp 2 terms
YesSame as question 2Except the signsare +-
Example
xsup2 - 64
NO proceed to question 4
( x - 8 ) ( x + 8)
Answer looks like this
4 No perfect squares on end 3 terms amp starts with xsup2
NO proceed to question 5
YES its quadratics in the morning
xsup2 - 10x + 24MA
Multiply to +24 Add to -10
38 11
-3 -8 -11
-2 -12 -14
-6-4 -10 Found it
Put in the parenthesis
(x-6) (x-4)
All Done
2 Search and Seizure (quadratics in the morning)(See question 4)
1 Steal the ldquoardquo and give it to the last term (Multiply)1
5 Is there a number in front of the xsup2 amp does it have 3 terms
Yes Jail BreakNO
Then it is Prime
(canrsquot factor)
Example 2xsup2 + 7x + 3
xsup2 + 7x + 6 (23)
( x + 6 ) ( x + 1 )
3 Arrested and Caught - Divide last terms by ldquoardquo
( x + 62 ) ( x + 12 )
4 Beat it Down - (reduce fractions)
( x + 3 ) ( x + 12 )
5 Parole (kick denominator to the front)(x + 3 ) ( 2 x + 1 )
All Done
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Check This First
Is there a common factor (number or letter)
Greatest Common Factor (GCF)
1 Factor each term2 Circle common terms3 Multiply common terms4 Write it down5 Put out baggie for leftover6 Multiply leftovers for each
term7 Put in baggie ( )
Example
8xsup2y + 4xsup3 - 12x
4x ( 2xy + xsup2 - 3 )
Is there a Square on each End
Perfect Squares
1 Put out Parenthesis2 Split FIRST and LAST numbers nice3 Put in Signs
3 Different Kinds
A xsup2 - 16x + 64
(xndash 8) (x ndash 8)
B xsup2 + 18x + 81
(x + 9) (x + 9)
C xsup2 - 36
(x + 6) (x ndash 6)
Is there a number in front of the xsup2 and does it have 3 terms
Jail Break
1Steal the ldquoardquo and give it to the last term (multiply)2Search and Seizure (Quad in AM)3Arrested and Caught (divide by ldquoardquo)4Beat Down (reduce fractions)5Parole (kick denominator to the front)6Check it out (FACE it)
Example
2xsup2 + 7x + 3
1xsup2 + 7x + 62(x + 6) (x + 1)3(x + 62) (x + 12)4(x + 3) (x +12)5(x + 3) (2x + 1)
No perfect squares and 3 terms and starts with xsup2
Quadratics in the Morning (AM)
1 Make a factor tree2 Multiply to last number add
to middle number3 Put out baggies (parenthesis)4 Split first term nice5 Drop in factors from tree
Example
xsup2 + 12x + 32
(x + 8) (x + 4)
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Solve with Graphing Calculator
1) Solve each equation for y (3 easy steps)
2) Use y= button and enter each equation
3) Use graph to eyeball answerOr
4) Use to find where y1 and y2 are equal
5) Be sure answer is in (x y) form
Y=
Systems of EquationsTo Solve by Graphing
1) Make an x y Chart
2) Select any x solve for y x 2x ndash 4 y 0 2(0) ndash 4 -4 1 2(1) ndash 4 -2
3) Then graph the two points4) Do for both equations5) The answer is where they
cross6) Be sure answer is in (x y)
form
2nd TABLE
bull y + 2x = 9
bull y = 9 ndash 2xbull 3(9 - 2x) ndash 2x = 11
bull 27-6x ndash 2x = 11bull 27 ndash 8x = 11bull - 8x = -16bull x = 2bull y = 9 ndash 2 (2)bull y = 5bull (2 5)
Systems of EquationsSolve by Substitution
(box amp shove)
1) Solve one equation for x or y (change sides change signs)
2) Box it3) Rewrite other equation
and shove box in 4) Solve for surviving letter
1) Distribute2) Combine like terms3) Solve
4) Send it back to box5) Solve for other letter6) Answer in (xy) form
3y -2x = 11y + 2x = 9
Systems of EquationsSolve by Elimination
1) Look for opposite signs
2) Multiply to create opposites
3) Add old and new equation together
4) Solve for surviving letter
5) Plug back into either equation (pick the easy one)
6) Solve for other letter
7) Answer in (xy) format
bull 2x ndash y = 93x + 4y = -14
bull 4(2x ndash y = 9)
bull 8x ndash 4y = 36bull 3x + 4y = -14bull 11x = 22bull x = 2
bull 2(2) ndash y = 9bull -y = 5bull y = -5
bull (2 -5)
2x ndash y = 93x + 4y = -14
Find Equation of the Line y = mx + b
I need slope (m) amp the y-intercept
(b)
MY ANSWER
y = x +
To find m ndash Solve the equation for y and use mor use the y2 ndash y1
x2 ndash x1 formula
To find b - Plug x y and m into the line equation and solve for b
(Just follow the Rules)
Base xsup2Exponent
Like Bases Exponent Example
Multiply Add
Power up Multiply
Divide Subtract s
Add No change 3asup2 + 5asup2 = 8asup2 asup2 + = asup2 +
Subtract No change 10asup2 - 4asup2 = 6asup2
5a
4
15 9 3a g y
6 4 5
3 7 4a x ka x k
5a
5a
5a
Exponents
=
(xsup2asup3g)(xasup2gsup3) = (xsup3 g )
( gsup3 y) sup3 =
asup3k xsup3
Negative ExponentsJust switch places and make exponent positive
5x5
1x
Example Switch Simplify
5 4 3
2 2 4x g ka x k
4 2
2 5 4 3g x
a x k k
4
2 3 7g
a x k
=
Xsup2
X X∙ = Xsup2 X
X times X is X squared
X
QuadraticsMultiplying Binomials ndash Draw the Face
(x + 8) (x ndash 6)
Multiply Watch your Signs
xsup2 +8x -6x -48
Simplify (Combine Like Terms eat the buggers)xsup2 + 2x -48
Check This First
The Math ldquoFrdquo Word ldquoFactoringrdquo
1 Is there a common factor (number or letter)
NO
Proceed to question 2
Yes
Proceed with CGF
4xsup3 2 2 x x x
4x ( 2xy + xsup2 -3 )
Example
8xsup2y + 4xsup3 - 12x
Factor each term8xsup2y 2 2 2 x x y 12x 2 2 3 x
Circle common terms
8xsup2y 2 2 2 x x y 4xsup3 2 2 x x x 12x 2 2 3 x
Multiply circled numbersThatrsquos your Common Factor
Multiply leftovers put in ( )
2 Perfect squares on end amp 3 terms
xsup2 - 10x + 25
YES SPLIT IT NICE
NO proceed to question 3
xsup2 - 10x + 25Put out baggies to hold the answer (parenthesis)
( x - 5 ) ( x - 5 )
Split the first term nice
Place in baggies
Sign between is same as middle term
Split the second term nice
Place in baggies
ANSWER
(x -5)sup2
3 Perfect squares on end amp 2 terms
YesSame as question 2Except the signsare +-
Example
xsup2 - 64
NO proceed to question 4
( x - 8 ) ( x + 8)
Answer looks like this
4 No perfect squares on end 3 terms amp starts with xsup2
NO proceed to question 5
YES its quadratics in the morning
xsup2 - 10x + 24MA
Multiply to +24 Add to -10
38 11
-3 -8 -11
-2 -12 -14
-6-4 -10 Found it
Put in the parenthesis
(x-6) (x-4)
All Done
2 Search and Seizure (quadratics in the morning)(See question 4)
1 Steal the ldquoardquo and give it to the last term (Multiply)1
5 Is there a number in front of the xsup2 amp does it have 3 terms
Yes Jail BreakNO
Then it is Prime
(canrsquot factor)
Example 2xsup2 + 7x + 3
xsup2 + 7x + 6 (23)
( x + 6 ) ( x + 1 )
3 Arrested and Caught - Divide last terms by ldquoardquo
( x + 62 ) ( x + 12 )
4 Beat it Down - (reduce fractions)
( x + 3 ) ( x + 12 )
5 Parole (kick denominator to the front)(x + 3 ) ( 2 x + 1 )
All Done
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Check This First
Is there a common factor (number or letter)
Greatest Common Factor (GCF)
1 Factor each term2 Circle common terms3 Multiply common terms4 Write it down5 Put out baggie for leftover6 Multiply leftovers for each
term7 Put in baggie ( )
Example
8xsup2y + 4xsup3 - 12x
4x ( 2xy + xsup2 - 3 )
Is there a Square on each End
Perfect Squares
1 Put out Parenthesis2 Split FIRST and LAST numbers nice3 Put in Signs
3 Different Kinds
A xsup2 - 16x + 64
(xndash 8) (x ndash 8)
B xsup2 + 18x + 81
(x + 9) (x + 9)
C xsup2 - 36
(x + 6) (x ndash 6)
Is there a number in front of the xsup2 and does it have 3 terms
Jail Break
1Steal the ldquoardquo and give it to the last term (multiply)2Search and Seizure (Quad in AM)3Arrested and Caught (divide by ldquoardquo)4Beat Down (reduce fractions)5Parole (kick denominator to the front)6Check it out (FACE it)
Example
2xsup2 + 7x + 3
1xsup2 + 7x + 62(x + 6) (x + 1)3(x + 62) (x + 12)4(x + 3) (x +12)5(x + 3) (2x + 1)
No perfect squares and 3 terms and starts with xsup2
Quadratics in the Morning (AM)
1 Make a factor tree2 Multiply to last number add
to middle number3 Put out baggies (parenthesis)4 Split first term nice5 Drop in factors from tree
Example
xsup2 + 12x + 32
(x + 8) (x + 4)
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Solve with Graphing Calculator
1) Solve each equation for y (3 easy steps)
2) Use y= button and enter each equation
3) Use graph to eyeball answerOr
4) Use to find where y1 and y2 are equal
5) Be sure answer is in (x y) form
Y=
Systems of EquationsTo Solve by Graphing
1) Make an x y Chart
2) Select any x solve for y x 2x ndash 4 y 0 2(0) ndash 4 -4 1 2(1) ndash 4 -2
3) Then graph the two points4) Do for both equations5) The answer is where they
cross6) Be sure answer is in (x y)
form
2nd TABLE
bull y + 2x = 9
bull y = 9 ndash 2xbull 3(9 - 2x) ndash 2x = 11
bull 27-6x ndash 2x = 11bull 27 ndash 8x = 11bull - 8x = -16bull x = 2bull y = 9 ndash 2 (2)bull y = 5bull (2 5)
Systems of EquationsSolve by Substitution
(box amp shove)
1) Solve one equation for x or y (change sides change signs)
2) Box it3) Rewrite other equation
and shove box in 4) Solve for surviving letter
1) Distribute2) Combine like terms3) Solve
4) Send it back to box5) Solve for other letter6) Answer in (xy) form
3y -2x = 11y + 2x = 9
Systems of EquationsSolve by Elimination
1) Look for opposite signs
2) Multiply to create opposites
3) Add old and new equation together
4) Solve for surviving letter
5) Plug back into either equation (pick the easy one)
6) Solve for other letter
7) Answer in (xy) format
bull 2x ndash y = 93x + 4y = -14
bull 4(2x ndash y = 9)
bull 8x ndash 4y = 36bull 3x + 4y = -14bull 11x = 22bull x = 2
bull 2(2) ndash y = 9bull -y = 5bull y = -5
bull (2 -5)
2x ndash y = 93x + 4y = -14
(Just follow the Rules)
Base xsup2Exponent
Like Bases Exponent Example
Multiply Add
Power up Multiply
Divide Subtract s
Add No change 3asup2 + 5asup2 = 8asup2 asup2 + = asup2 +
Subtract No change 10asup2 - 4asup2 = 6asup2
5a
4
15 9 3a g y
6 4 5
3 7 4a x ka x k
5a
5a
5a
Exponents
=
(xsup2asup3g)(xasup2gsup3) = (xsup3 g )
( gsup3 y) sup3 =
asup3k xsup3
Negative ExponentsJust switch places and make exponent positive
5x5
1x
Example Switch Simplify
5 4 3
2 2 4x g ka x k
4 2
2 5 4 3g x
a x k k
4
2 3 7g
a x k
=
Xsup2
X X∙ = Xsup2 X
X times X is X squared
X
QuadraticsMultiplying Binomials ndash Draw the Face
(x + 8) (x ndash 6)
Multiply Watch your Signs
xsup2 +8x -6x -48
Simplify (Combine Like Terms eat the buggers)xsup2 + 2x -48
Check This First
The Math ldquoFrdquo Word ldquoFactoringrdquo
1 Is there a common factor (number or letter)
NO
Proceed to question 2
Yes
Proceed with CGF
4xsup3 2 2 x x x
4x ( 2xy + xsup2 -3 )
Example
8xsup2y + 4xsup3 - 12x
Factor each term8xsup2y 2 2 2 x x y 12x 2 2 3 x
Circle common terms
8xsup2y 2 2 2 x x y 4xsup3 2 2 x x x 12x 2 2 3 x
Multiply circled numbersThatrsquos your Common Factor
Multiply leftovers put in ( )
2 Perfect squares on end amp 3 terms
xsup2 - 10x + 25
YES SPLIT IT NICE
NO proceed to question 3
xsup2 - 10x + 25Put out baggies to hold the answer (parenthesis)
( x - 5 ) ( x - 5 )
Split the first term nice
Place in baggies
Sign between is same as middle term
Split the second term nice
Place in baggies
ANSWER
(x -5)sup2
3 Perfect squares on end amp 2 terms
YesSame as question 2Except the signsare +-
Example
xsup2 - 64
NO proceed to question 4
( x - 8 ) ( x + 8)
Answer looks like this
4 No perfect squares on end 3 terms amp starts with xsup2
NO proceed to question 5
YES its quadratics in the morning
xsup2 - 10x + 24MA
Multiply to +24 Add to -10
38 11
-3 -8 -11
-2 -12 -14
-6-4 -10 Found it
Put in the parenthesis
(x-6) (x-4)
All Done
2 Search and Seizure (quadratics in the morning)(See question 4)
1 Steal the ldquoardquo and give it to the last term (Multiply)1
5 Is there a number in front of the xsup2 amp does it have 3 terms
Yes Jail BreakNO
Then it is Prime
(canrsquot factor)
Example 2xsup2 + 7x + 3
xsup2 + 7x + 6 (23)
( x + 6 ) ( x + 1 )
3 Arrested and Caught - Divide last terms by ldquoardquo
( x + 62 ) ( x + 12 )
4 Beat it Down - (reduce fractions)
( x + 3 ) ( x + 12 )
5 Parole (kick denominator to the front)(x + 3 ) ( 2 x + 1 )
All Done
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Check This First
Is there a common factor (number or letter)
Greatest Common Factor (GCF)
1 Factor each term2 Circle common terms3 Multiply common terms4 Write it down5 Put out baggie for leftover6 Multiply leftovers for each
term7 Put in baggie ( )
Example
8xsup2y + 4xsup3 - 12x
4x ( 2xy + xsup2 - 3 )
Is there a Square on each End
Perfect Squares
1 Put out Parenthesis2 Split FIRST and LAST numbers nice3 Put in Signs
3 Different Kinds
A xsup2 - 16x + 64
(xndash 8) (x ndash 8)
B xsup2 + 18x + 81
(x + 9) (x + 9)
C xsup2 - 36
(x + 6) (x ndash 6)
Is there a number in front of the xsup2 and does it have 3 terms
Jail Break
1Steal the ldquoardquo and give it to the last term (multiply)2Search and Seizure (Quad in AM)3Arrested and Caught (divide by ldquoardquo)4Beat Down (reduce fractions)5Parole (kick denominator to the front)6Check it out (FACE it)
Example
2xsup2 + 7x + 3
1xsup2 + 7x + 62(x + 6) (x + 1)3(x + 62) (x + 12)4(x + 3) (x +12)5(x + 3) (2x + 1)
No perfect squares and 3 terms and starts with xsup2
Quadratics in the Morning (AM)
1 Make a factor tree2 Multiply to last number add
to middle number3 Put out baggies (parenthesis)4 Split first term nice5 Drop in factors from tree
Example
xsup2 + 12x + 32
(x + 8) (x + 4)
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Solve with Graphing Calculator
1) Solve each equation for y (3 easy steps)
2) Use y= button and enter each equation
3) Use graph to eyeball answerOr
4) Use to find where y1 and y2 are equal
5) Be sure answer is in (x y) form
Y=
Systems of EquationsTo Solve by Graphing
1) Make an x y Chart
2) Select any x solve for y x 2x ndash 4 y 0 2(0) ndash 4 -4 1 2(1) ndash 4 -2
3) Then graph the two points4) Do for both equations5) The answer is where they
cross6) Be sure answer is in (x y)
form
2nd TABLE
bull y + 2x = 9
bull y = 9 ndash 2xbull 3(9 - 2x) ndash 2x = 11
bull 27-6x ndash 2x = 11bull 27 ndash 8x = 11bull - 8x = -16bull x = 2bull y = 9 ndash 2 (2)bull y = 5bull (2 5)
Systems of EquationsSolve by Substitution
(box amp shove)
1) Solve one equation for x or y (change sides change signs)
2) Box it3) Rewrite other equation
and shove box in 4) Solve for surviving letter
1) Distribute2) Combine like terms3) Solve
4) Send it back to box5) Solve for other letter6) Answer in (xy) form
3y -2x = 11y + 2x = 9
Systems of EquationsSolve by Elimination
1) Look for opposite signs
2) Multiply to create opposites
3) Add old and new equation together
4) Solve for surviving letter
5) Plug back into either equation (pick the easy one)
6) Solve for other letter
7) Answer in (xy) format
bull 2x ndash y = 93x + 4y = -14
bull 4(2x ndash y = 9)
bull 8x ndash 4y = 36bull 3x + 4y = -14bull 11x = 22bull x = 2
bull 2(2) ndash y = 9bull -y = 5bull y = -5
bull (2 -5)
2x ndash y = 93x + 4y = -14
Negative ExponentsJust switch places and make exponent positive
5x5
1x
Example Switch Simplify
5 4 3
2 2 4x g ka x k
4 2
2 5 4 3g x
a x k k
4
2 3 7g
a x k
=
Xsup2
X X∙ = Xsup2 X
X times X is X squared
X
QuadraticsMultiplying Binomials ndash Draw the Face
(x + 8) (x ndash 6)
Multiply Watch your Signs
xsup2 +8x -6x -48
Simplify (Combine Like Terms eat the buggers)xsup2 + 2x -48
Check This First
The Math ldquoFrdquo Word ldquoFactoringrdquo
1 Is there a common factor (number or letter)
NO
Proceed to question 2
Yes
Proceed with CGF
4xsup3 2 2 x x x
4x ( 2xy + xsup2 -3 )
Example
8xsup2y + 4xsup3 - 12x
Factor each term8xsup2y 2 2 2 x x y 12x 2 2 3 x
Circle common terms
8xsup2y 2 2 2 x x y 4xsup3 2 2 x x x 12x 2 2 3 x
Multiply circled numbersThatrsquos your Common Factor
Multiply leftovers put in ( )
2 Perfect squares on end amp 3 terms
xsup2 - 10x + 25
YES SPLIT IT NICE
NO proceed to question 3
xsup2 - 10x + 25Put out baggies to hold the answer (parenthesis)
( x - 5 ) ( x - 5 )
Split the first term nice
Place in baggies
Sign between is same as middle term
Split the second term nice
Place in baggies
ANSWER
(x -5)sup2
3 Perfect squares on end amp 2 terms
YesSame as question 2Except the signsare +-
Example
xsup2 - 64
NO proceed to question 4
( x - 8 ) ( x + 8)
Answer looks like this
4 No perfect squares on end 3 terms amp starts with xsup2
NO proceed to question 5
YES its quadratics in the morning
xsup2 - 10x + 24MA
Multiply to +24 Add to -10
38 11
-3 -8 -11
-2 -12 -14
-6-4 -10 Found it
Put in the parenthesis
(x-6) (x-4)
All Done
2 Search and Seizure (quadratics in the morning)(See question 4)
1 Steal the ldquoardquo and give it to the last term (Multiply)1
5 Is there a number in front of the xsup2 amp does it have 3 terms
Yes Jail BreakNO
Then it is Prime
(canrsquot factor)
Example 2xsup2 + 7x + 3
xsup2 + 7x + 6 (23)
( x + 6 ) ( x + 1 )
3 Arrested and Caught - Divide last terms by ldquoardquo
( x + 62 ) ( x + 12 )
4 Beat it Down - (reduce fractions)
( x + 3 ) ( x + 12 )
5 Parole (kick denominator to the front)(x + 3 ) ( 2 x + 1 )
All Done
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Check This First
Is there a common factor (number or letter)
Greatest Common Factor (GCF)
1 Factor each term2 Circle common terms3 Multiply common terms4 Write it down5 Put out baggie for leftover6 Multiply leftovers for each
term7 Put in baggie ( )
Example
8xsup2y + 4xsup3 - 12x
4x ( 2xy + xsup2 - 3 )
Is there a Square on each End
Perfect Squares
1 Put out Parenthesis2 Split FIRST and LAST numbers nice3 Put in Signs
3 Different Kinds
A xsup2 - 16x + 64
(xndash 8) (x ndash 8)
B xsup2 + 18x + 81
(x + 9) (x + 9)
C xsup2 - 36
(x + 6) (x ndash 6)
Is there a number in front of the xsup2 and does it have 3 terms
Jail Break
1Steal the ldquoardquo and give it to the last term (multiply)2Search and Seizure (Quad in AM)3Arrested and Caught (divide by ldquoardquo)4Beat Down (reduce fractions)5Parole (kick denominator to the front)6Check it out (FACE it)
Example
2xsup2 + 7x + 3
1xsup2 + 7x + 62(x + 6) (x + 1)3(x + 62) (x + 12)4(x + 3) (x +12)5(x + 3) (2x + 1)
No perfect squares and 3 terms and starts with xsup2
Quadratics in the Morning (AM)
1 Make a factor tree2 Multiply to last number add
to middle number3 Put out baggies (parenthesis)4 Split first term nice5 Drop in factors from tree
Example
xsup2 + 12x + 32
(x + 8) (x + 4)
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Solve with Graphing Calculator
1) Solve each equation for y (3 easy steps)
2) Use y= button and enter each equation
3) Use graph to eyeball answerOr
4) Use to find where y1 and y2 are equal
5) Be sure answer is in (x y) form
Y=
Systems of EquationsTo Solve by Graphing
1) Make an x y Chart
2) Select any x solve for y x 2x ndash 4 y 0 2(0) ndash 4 -4 1 2(1) ndash 4 -2
3) Then graph the two points4) Do for both equations5) The answer is where they
cross6) Be sure answer is in (x y)
form
2nd TABLE
bull y + 2x = 9
bull y = 9 ndash 2xbull 3(9 - 2x) ndash 2x = 11
bull 27-6x ndash 2x = 11bull 27 ndash 8x = 11bull - 8x = -16bull x = 2bull y = 9 ndash 2 (2)bull y = 5bull (2 5)
Systems of EquationsSolve by Substitution
(box amp shove)
1) Solve one equation for x or y (change sides change signs)
2) Box it3) Rewrite other equation
and shove box in 4) Solve for surviving letter
1) Distribute2) Combine like terms3) Solve
4) Send it back to box5) Solve for other letter6) Answer in (xy) form
3y -2x = 11y + 2x = 9
Systems of EquationsSolve by Elimination
1) Look for opposite signs
2) Multiply to create opposites
3) Add old and new equation together
4) Solve for surviving letter
5) Plug back into either equation (pick the easy one)
6) Solve for other letter
7) Answer in (xy) format
bull 2x ndash y = 93x + 4y = -14
bull 4(2x ndash y = 9)
bull 8x ndash 4y = 36bull 3x + 4y = -14bull 11x = 22bull x = 2
bull 2(2) ndash y = 9bull -y = 5bull y = -5
bull (2 -5)
2x ndash y = 93x + 4y = -14
Xsup2
X X∙ = Xsup2 X
X times X is X squared
X
QuadraticsMultiplying Binomials ndash Draw the Face
(x + 8) (x ndash 6)
Multiply Watch your Signs
xsup2 +8x -6x -48
Simplify (Combine Like Terms eat the buggers)xsup2 + 2x -48
Check This First
The Math ldquoFrdquo Word ldquoFactoringrdquo
1 Is there a common factor (number or letter)
NO
Proceed to question 2
Yes
Proceed with CGF
4xsup3 2 2 x x x
4x ( 2xy + xsup2 -3 )
Example
8xsup2y + 4xsup3 - 12x
Factor each term8xsup2y 2 2 2 x x y 12x 2 2 3 x
Circle common terms
8xsup2y 2 2 2 x x y 4xsup3 2 2 x x x 12x 2 2 3 x
Multiply circled numbersThatrsquos your Common Factor
Multiply leftovers put in ( )
2 Perfect squares on end amp 3 terms
xsup2 - 10x + 25
YES SPLIT IT NICE
NO proceed to question 3
xsup2 - 10x + 25Put out baggies to hold the answer (parenthesis)
( x - 5 ) ( x - 5 )
Split the first term nice
Place in baggies
Sign between is same as middle term
Split the second term nice
Place in baggies
ANSWER
(x -5)sup2
3 Perfect squares on end amp 2 terms
YesSame as question 2Except the signsare +-
Example
xsup2 - 64
NO proceed to question 4
( x - 8 ) ( x + 8)
Answer looks like this
4 No perfect squares on end 3 terms amp starts with xsup2
NO proceed to question 5
YES its quadratics in the morning
xsup2 - 10x + 24MA
Multiply to +24 Add to -10
38 11
-3 -8 -11
-2 -12 -14
-6-4 -10 Found it
Put in the parenthesis
(x-6) (x-4)
All Done
2 Search and Seizure (quadratics in the morning)(See question 4)
1 Steal the ldquoardquo and give it to the last term (Multiply)1
5 Is there a number in front of the xsup2 amp does it have 3 terms
Yes Jail BreakNO
Then it is Prime
(canrsquot factor)
Example 2xsup2 + 7x + 3
xsup2 + 7x + 6 (23)
( x + 6 ) ( x + 1 )
3 Arrested and Caught - Divide last terms by ldquoardquo
( x + 62 ) ( x + 12 )
4 Beat it Down - (reduce fractions)
( x + 3 ) ( x + 12 )
5 Parole (kick denominator to the front)(x + 3 ) ( 2 x + 1 )
All Done
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Check This First
Is there a common factor (number or letter)
Greatest Common Factor (GCF)
1 Factor each term2 Circle common terms3 Multiply common terms4 Write it down5 Put out baggie for leftover6 Multiply leftovers for each
term7 Put in baggie ( )
Example
8xsup2y + 4xsup3 - 12x
4x ( 2xy + xsup2 - 3 )
Is there a Square on each End
Perfect Squares
1 Put out Parenthesis2 Split FIRST and LAST numbers nice3 Put in Signs
3 Different Kinds
A xsup2 - 16x + 64
(xndash 8) (x ndash 8)
B xsup2 + 18x + 81
(x + 9) (x + 9)
C xsup2 - 36
(x + 6) (x ndash 6)
Is there a number in front of the xsup2 and does it have 3 terms
Jail Break
1Steal the ldquoardquo and give it to the last term (multiply)2Search and Seizure (Quad in AM)3Arrested and Caught (divide by ldquoardquo)4Beat Down (reduce fractions)5Parole (kick denominator to the front)6Check it out (FACE it)
Example
2xsup2 + 7x + 3
1xsup2 + 7x + 62(x + 6) (x + 1)3(x + 62) (x + 12)4(x + 3) (x +12)5(x + 3) (2x + 1)
No perfect squares and 3 terms and starts with xsup2
Quadratics in the Morning (AM)
1 Make a factor tree2 Multiply to last number add
to middle number3 Put out baggies (parenthesis)4 Split first term nice5 Drop in factors from tree
Example
xsup2 + 12x + 32
(x + 8) (x + 4)
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Solve with Graphing Calculator
1) Solve each equation for y (3 easy steps)
2) Use y= button and enter each equation
3) Use graph to eyeball answerOr
4) Use to find where y1 and y2 are equal
5) Be sure answer is in (x y) form
Y=
Systems of EquationsTo Solve by Graphing
1) Make an x y Chart
2) Select any x solve for y x 2x ndash 4 y 0 2(0) ndash 4 -4 1 2(1) ndash 4 -2
3) Then graph the two points4) Do for both equations5) The answer is where they
cross6) Be sure answer is in (x y)
form
2nd TABLE
bull y + 2x = 9
bull y = 9 ndash 2xbull 3(9 - 2x) ndash 2x = 11
bull 27-6x ndash 2x = 11bull 27 ndash 8x = 11bull - 8x = -16bull x = 2bull y = 9 ndash 2 (2)bull y = 5bull (2 5)
Systems of EquationsSolve by Substitution
(box amp shove)
1) Solve one equation for x or y (change sides change signs)
2) Box it3) Rewrite other equation
and shove box in 4) Solve for surviving letter
1) Distribute2) Combine like terms3) Solve
4) Send it back to box5) Solve for other letter6) Answer in (xy) form
3y -2x = 11y + 2x = 9
Systems of EquationsSolve by Elimination
1) Look for opposite signs
2) Multiply to create opposites
3) Add old and new equation together
4) Solve for surviving letter
5) Plug back into either equation (pick the easy one)
6) Solve for other letter
7) Answer in (xy) format
bull 2x ndash y = 93x + 4y = -14
bull 4(2x ndash y = 9)
bull 8x ndash 4y = 36bull 3x + 4y = -14bull 11x = 22bull x = 2
bull 2(2) ndash y = 9bull -y = 5bull y = -5
bull (2 -5)
2x ndash y = 93x + 4y = -14
QuadraticsMultiplying Binomials ndash Draw the Face
(x + 8) (x ndash 6)
Multiply Watch your Signs
xsup2 +8x -6x -48
Simplify (Combine Like Terms eat the buggers)xsup2 + 2x -48
Check This First
The Math ldquoFrdquo Word ldquoFactoringrdquo
1 Is there a common factor (number or letter)
NO
Proceed to question 2
Yes
Proceed with CGF
4xsup3 2 2 x x x
4x ( 2xy + xsup2 -3 )
Example
8xsup2y + 4xsup3 - 12x
Factor each term8xsup2y 2 2 2 x x y 12x 2 2 3 x
Circle common terms
8xsup2y 2 2 2 x x y 4xsup3 2 2 x x x 12x 2 2 3 x
Multiply circled numbersThatrsquos your Common Factor
Multiply leftovers put in ( )
2 Perfect squares on end amp 3 terms
xsup2 - 10x + 25
YES SPLIT IT NICE
NO proceed to question 3
xsup2 - 10x + 25Put out baggies to hold the answer (parenthesis)
( x - 5 ) ( x - 5 )
Split the first term nice
Place in baggies
Sign between is same as middle term
Split the second term nice
Place in baggies
ANSWER
(x -5)sup2
3 Perfect squares on end amp 2 terms
YesSame as question 2Except the signsare +-
Example
xsup2 - 64
NO proceed to question 4
( x - 8 ) ( x + 8)
Answer looks like this
4 No perfect squares on end 3 terms amp starts with xsup2
NO proceed to question 5
YES its quadratics in the morning
xsup2 - 10x + 24MA
Multiply to +24 Add to -10
38 11
-3 -8 -11
-2 -12 -14
-6-4 -10 Found it
Put in the parenthesis
(x-6) (x-4)
All Done
2 Search and Seizure (quadratics in the morning)(See question 4)
1 Steal the ldquoardquo and give it to the last term (Multiply)1
5 Is there a number in front of the xsup2 amp does it have 3 terms
Yes Jail BreakNO
Then it is Prime
(canrsquot factor)
Example 2xsup2 + 7x + 3
xsup2 + 7x + 6 (23)
( x + 6 ) ( x + 1 )
3 Arrested and Caught - Divide last terms by ldquoardquo
( x + 62 ) ( x + 12 )
4 Beat it Down - (reduce fractions)
( x + 3 ) ( x + 12 )
5 Parole (kick denominator to the front)(x + 3 ) ( 2 x + 1 )
All Done
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Check This First
Is there a common factor (number or letter)
Greatest Common Factor (GCF)
1 Factor each term2 Circle common terms3 Multiply common terms4 Write it down5 Put out baggie for leftover6 Multiply leftovers for each
term7 Put in baggie ( )
Example
8xsup2y + 4xsup3 - 12x
4x ( 2xy + xsup2 - 3 )
Is there a Square on each End
Perfect Squares
1 Put out Parenthesis2 Split FIRST and LAST numbers nice3 Put in Signs
3 Different Kinds
A xsup2 - 16x + 64
(xndash 8) (x ndash 8)
B xsup2 + 18x + 81
(x + 9) (x + 9)
C xsup2 - 36
(x + 6) (x ndash 6)
Is there a number in front of the xsup2 and does it have 3 terms
Jail Break
1Steal the ldquoardquo and give it to the last term (multiply)2Search and Seizure (Quad in AM)3Arrested and Caught (divide by ldquoardquo)4Beat Down (reduce fractions)5Parole (kick denominator to the front)6Check it out (FACE it)
Example
2xsup2 + 7x + 3
1xsup2 + 7x + 62(x + 6) (x + 1)3(x + 62) (x + 12)4(x + 3) (x +12)5(x + 3) (2x + 1)
No perfect squares and 3 terms and starts with xsup2
Quadratics in the Morning (AM)
1 Make a factor tree2 Multiply to last number add
to middle number3 Put out baggies (parenthesis)4 Split first term nice5 Drop in factors from tree
Example
xsup2 + 12x + 32
(x + 8) (x + 4)
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Solve with Graphing Calculator
1) Solve each equation for y (3 easy steps)
2) Use y= button and enter each equation
3) Use graph to eyeball answerOr
4) Use to find where y1 and y2 are equal
5) Be sure answer is in (x y) form
Y=
Systems of EquationsTo Solve by Graphing
1) Make an x y Chart
2) Select any x solve for y x 2x ndash 4 y 0 2(0) ndash 4 -4 1 2(1) ndash 4 -2
3) Then graph the two points4) Do for both equations5) The answer is where they
cross6) Be sure answer is in (x y)
form
2nd TABLE
bull y + 2x = 9
bull y = 9 ndash 2xbull 3(9 - 2x) ndash 2x = 11
bull 27-6x ndash 2x = 11bull 27 ndash 8x = 11bull - 8x = -16bull x = 2bull y = 9 ndash 2 (2)bull y = 5bull (2 5)
Systems of EquationsSolve by Substitution
(box amp shove)
1) Solve one equation for x or y (change sides change signs)
2) Box it3) Rewrite other equation
and shove box in 4) Solve for surviving letter
1) Distribute2) Combine like terms3) Solve
4) Send it back to box5) Solve for other letter6) Answer in (xy) form
3y -2x = 11y + 2x = 9
Systems of EquationsSolve by Elimination
1) Look for opposite signs
2) Multiply to create opposites
3) Add old and new equation together
4) Solve for surviving letter
5) Plug back into either equation (pick the easy one)
6) Solve for other letter
7) Answer in (xy) format
bull 2x ndash y = 93x + 4y = -14
bull 4(2x ndash y = 9)
bull 8x ndash 4y = 36bull 3x + 4y = -14bull 11x = 22bull x = 2
bull 2(2) ndash y = 9bull -y = 5bull y = -5
bull (2 -5)
2x ndash y = 93x + 4y = -14
Check This First
The Math ldquoFrdquo Word ldquoFactoringrdquo
1 Is there a common factor (number or letter)
NO
Proceed to question 2
Yes
Proceed with CGF
4xsup3 2 2 x x x
4x ( 2xy + xsup2 -3 )
Example
8xsup2y + 4xsup3 - 12x
Factor each term8xsup2y 2 2 2 x x y 12x 2 2 3 x
Circle common terms
8xsup2y 2 2 2 x x y 4xsup3 2 2 x x x 12x 2 2 3 x
Multiply circled numbersThatrsquos your Common Factor
Multiply leftovers put in ( )
2 Perfect squares on end amp 3 terms
xsup2 - 10x + 25
YES SPLIT IT NICE
NO proceed to question 3
xsup2 - 10x + 25Put out baggies to hold the answer (parenthesis)
( x - 5 ) ( x - 5 )
Split the first term nice
Place in baggies
Sign between is same as middle term
Split the second term nice
Place in baggies
ANSWER
(x -5)sup2
3 Perfect squares on end amp 2 terms
YesSame as question 2Except the signsare +-
Example
xsup2 - 64
NO proceed to question 4
( x - 8 ) ( x + 8)
Answer looks like this
4 No perfect squares on end 3 terms amp starts with xsup2
NO proceed to question 5
YES its quadratics in the morning
xsup2 - 10x + 24MA
Multiply to +24 Add to -10
38 11
-3 -8 -11
-2 -12 -14
-6-4 -10 Found it
Put in the parenthesis
(x-6) (x-4)
All Done
2 Search and Seizure (quadratics in the morning)(See question 4)
1 Steal the ldquoardquo and give it to the last term (Multiply)1
5 Is there a number in front of the xsup2 amp does it have 3 terms
Yes Jail BreakNO
Then it is Prime
(canrsquot factor)
Example 2xsup2 + 7x + 3
xsup2 + 7x + 6 (23)
( x + 6 ) ( x + 1 )
3 Arrested and Caught - Divide last terms by ldquoardquo
( x + 62 ) ( x + 12 )
4 Beat it Down - (reduce fractions)
( x + 3 ) ( x + 12 )
5 Parole (kick denominator to the front)(x + 3 ) ( 2 x + 1 )
All Done
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Check This First
Is there a common factor (number or letter)
Greatest Common Factor (GCF)
1 Factor each term2 Circle common terms3 Multiply common terms4 Write it down5 Put out baggie for leftover6 Multiply leftovers for each
term7 Put in baggie ( )
Example
8xsup2y + 4xsup3 - 12x
4x ( 2xy + xsup2 - 3 )
Is there a Square on each End
Perfect Squares
1 Put out Parenthesis2 Split FIRST and LAST numbers nice3 Put in Signs
3 Different Kinds
A xsup2 - 16x + 64
(xndash 8) (x ndash 8)
B xsup2 + 18x + 81
(x + 9) (x + 9)
C xsup2 - 36
(x + 6) (x ndash 6)
Is there a number in front of the xsup2 and does it have 3 terms
Jail Break
1Steal the ldquoardquo and give it to the last term (multiply)2Search and Seizure (Quad in AM)3Arrested and Caught (divide by ldquoardquo)4Beat Down (reduce fractions)5Parole (kick denominator to the front)6Check it out (FACE it)
Example
2xsup2 + 7x + 3
1xsup2 + 7x + 62(x + 6) (x + 1)3(x + 62) (x + 12)4(x + 3) (x +12)5(x + 3) (2x + 1)
No perfect squares and 3 terms and starts with xsup2
Quadratics in the Morning (AM)
1 Make a factor tree2 Multiply to last number add
to middle number3 Put out baggies (parenthesis)4 Split first term nice5 Drop in factors from tree
Example
xsup2 + 12x + 32
(x + 8) (x + 4)
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Solve with Graphing Calculator
1) Solve each equation for y (3 easy steps)
2) Use y= button and enter each equation
3) Use graph to eyeball answerOr
4) Use to find where y1 and y2 are equal
5) Be sure answer is in (x y) form
Y=
Systems of EquationsTo Solve by Graphing
1) Make an x y Chart
2) Select any x solve for y x 2x ndash 4 y 0 2(0) ndash 4 -4 1 2(1) ndash 4 -2
3) Then graph the two points4) Do for both equations5) The answer is where they
cross6) Be sure answer is in (x y)
form
2nd TABLE
bull y + 2x = 9
bull y = 9 ndash 2xbull 3(9 - 2x) ndash 2x = 11
bull 27-6x ndash 2x = 11bull 27 ndash 8x = 11bull - 8x = -16bull x = 2bull y = 9 ndash 2 (2)bull y = 5bull (2 5)
Systems of EquationsSolve by Substitution
(box amp shove)
1) Solve one equation for x or y (change sides change signs)
2) Box it3) Rewrite other equation
and shove box in 4) Solve for surviving letter
1) Distribute2) Combine like terms3) Solve
4) Send it back to box5) Solve for other letter6) Answer in (xy) form
3y -2x = 11y + 2x = 9
Systems of EquationsSolve by Elimination
1) Look for opposite signs
2) Multiply to create opposites
3) Add old and new equation together
4) Solve for surviving letter
5) Plug back into either equation (pick the easy one)
6) Solve for other letter
7) Answer in (xy) format
bull 2x ndash y = 93x + 4y = -14
bull 4(2x ndash y = 9)
bull 8x ndash 4y = 36bull 3x + 4y = -14bull 11x = 22bull x = 2
bull 2(2) ndash y = 9bull -y = 5bull y = -5
bull (2 -5)
2x ndash y = 93x + 4y = -14
2 Perfect squares on end amp 3 terms
xsup2 - 10x + 25
YES SPLIT IT NICE
NO proceed to question 3
xsup2 - 10x + 25Put out baggies to hold the answer (parenthesis)
( x - 5 ) ( x - 5 )
Split the first term nice
Place in baggies
Sign between is same as middle term
Split the second term nice
Place in baggies
ANSWER
(x -5)sup2
3 Perfect squares on end amp 2 terms
YesSame as question 2Except the signsare +-
Example
xsup2 - 64
NO proceed to question 4
( x - 8 ) ( x + 8)
Answer looks like this
4 No perfect squares on end 3 terms amp starts with xsup2
NO proceed to question 5
YES its quadratics in the morning
xsup2 - 10x + 24MA
Multiply to +24 Add to -10
38 11
-3 -8 -11
-2 -12 -14
-6-4 -10 Found it
Put in the parenthesis
(x-6) (x-4)
All Done
2 Search and Seizure (quadratics in the morning)(See question 4)
1 Steal the ldquoardquo and give it to the last term (Multiply)1
5 Is there a number in front of the xsup2 amp does it have 3 terms
Yes Jail BreakNO
Then it is Prime
(canrsquot factor)
Example 2xsup2 + 7x + 3
xsup2 + 7x + 6 (23)
( x + 6 ) ( x + 1 )
3 Arrested and Caught - Divide last terms by ldquoardquo
( x + 62 ) ( x + 12 )
4 Beat it Down - (reduce fractions)
( x + 3 ) ( x + 12 )
5 Parole (kick denominator to the front)(x + 3 ) ( 2 x + 1 )
All Done
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Check This First
Is there a common factor (number or letter)
Greatest Common Factor (GCF)
1 Factor each term2 Circle common terms3 Multiply common terms4 Write it down5 Put out baggie for leftover6 Multiply leftovers for each
term7 Put in baggie ( )
Example
8xsup2y + 4xsup3 - 12x
4x ( 2xy + xsup2 - 3 )
Is there a Square on each End
Perfect Squares
1 Put out Parenthesis2 Split FIRST and LAST numbers nice3 Put in Signs
3 Different Kinds
A xsup2 - 16x + 64
(xndash 8) (x ndash 8)
B xsup2 + 18x + 81
(x + 9) (x + 9)
C xsup2 - 36
(x + 6) (x ndash 6)
Is there a number in front of the xsup2 and does it have 3 terms
Jail Break
1Steal the ldquoardquo and give it to the last term (multiply)2Search and Seizure (Quad in AM)3Arrested and Caught (divide by ldquoardquo)4Beat Down (reduce fractions)5Parole (kick denominator to the front)6Check it out (FACE it)
Example
2xsup2 + 7x + 3
1xsup2 + 7x + 62(x + 6) (x + 1)3(x + 62) (x + 12)4(x + 3) (x +12)5(x + 3) (2x + 1)
No perfect squares and 3 terms and starts with xsup2
Quadratics in the Morning (AM)
1 Make a factor tree2 Multiply to last number add
to middle number3 Put out baggies (parenthesis)4 Split first term nice5 Drop in factors from tree
Example
xsup2 + 12x + 32
(x + 8) (x + 4)
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Solve with Graphing Calculator
1) Solve each equation for y (3 easy steps)
2) Use y= button and enter each equation
3) Use graph to eyeball answerOr
4) Use to find where y1 and y2 are equal
5) Be sure answer is in (x y) form
Y=
Systems of EquationsTo Solve by Graphing
1) Make an x y Chart
2) Select any x solve for y x 2x ndash 4 y 0 2(0) ndash 4 -4 1 2(1) ndash 4 -2
3) Then graph the two points4) Do for both equations5) The answer is where they
cross6) Be sure answer is in (x y)
form
2nd TABLE
bull y + 2x = 9
bull y = 9 ndash 2xbull 3(9 - 2x) ndash 2x = 11
bull 27-6x ndash 2x = 11bull 27 ndash 8x = 11bull - 8x = -16bull x = 2bull y = 9 ndash 2 (2)bull y = 5bull (2 5)
Systems of EquationsSolve by Substitution
(box amp shove)
1) Solve one equation for x or y (change sides change signs)
2) Box it3) Rewrite other equation
and shove box in 4) Solve for surviving letter
1) Distribute2) Combine like terms3) Solve
4) Send it back to box5) Solve for other letter6) Answer in (xy) form
3y -2x = 11y + 2x = 9
Systems of EquationsSolve by Elimination
1) Look for opposite signs
2) Multiply to create opposites
3) Add old and new equation together
4) Solve for surviving letter
5) Plug back into either equation (pick the easy one)
6) Solve for other letter
7) Answer in (xy) format
bull 2x ndash y = 93x + 4y = -14
bull 4(2x ndash y = 9)
bull 8x ndash 4y = 36bull 3x + 4y = -14bull 11x = 22bull x = 2
bull 2(2) ndash y = 9bull -y = 5bull y = -5
bull (2 -5)
2x ndash y = 93x + 4y = -14
3 Perfect squares on end amp 2 terms
YesSame as question 2Except the signsare +-
Example
xsup2 - 64
NO proceed to question 4
( x - 8 ) ( x + 8)
Answer looks like this
4 No perfect squares on end 3 terms amp starts with xsup2
NO proceed to question 5
YES its quadratics in the morning
xsup2 - 10x + 24MA
Multiply to +24 Add to -10
38 11
-3 -8 -11
-2 -12 -14
-6-4 -10 Found it
Put in the parenthesis
(x-6) (x-4)
All Done
2 Search and Seizure (quadratics in the morning)(See question 4)
1 Steal the ldquoardquo and give it to the last term (Multiply)1
5 Is there a number in front of the xsup2 amp does it have 3 terms
Yes Jail BreakNO
Then it is Prime
(canrsquot factor)
Example 2xsup2 + 7x + 3
xsup2 + 7x + 6 (23)
( x + 6 ) ( x + 1 )
3 Arrested and Caught - Divide last terms by ldquoardquo
( x + 62 ) ( x + 12 )
4 Beat it Down - (reduce fractions)
( x + 3 ) ( x + 12 )
5 Parole (kick denominator to the front)(x + 3 ) ( 2 x + 1 )
All Done
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Check This First
Is there a common factor (number or letter)
Greatest Common Factor (GCF)
1 Factor each term2 Circle common terms3 Multiply common terms4 Write it down5 Put out baggie for leftover6 Multiply leftovers for each
term7 Put in baggie ( )
Example
8xsup2y + 4xsup3 - 12x
4x ( 2xy + xsup2 - 3 )
Is there a Square on each End
Perfect Squares
1 Put out Parenthesis2 Split FIRST and LAST numbers nice3 Put in Signs
3 Different Kinds
A xsup2 - 16x + 64
(xndash 8) (x ndash 8)
B xsup2 + 18x + 81
(x + 9) (x + 9)
C xsup2 - 36
(x + 6) (x ndash 6)
Is there a number in front of the xsup2 and does it have 3 terms
Jail Break
1Steal the ldquoardquo and give it to the last term (multiply)2Search and Seizure (Quad in AM)3Arrested and Caught (divide by ldquoardquo)4Beat Down (reduce fractions)5Parole (kick denominator to the front)6Check it out (FACE it)
Example
2xsup2 + 7x + 3
1xsup2 + 7x + 62(x + 6) (x + 1)3(x + 62) (x + 12)4(x + 3) (x +12)5(x + 3) (2x + 1)
No perfect squares and 3 terms and starts with xsup2
Quadratics in the Morning (AM)
1 Make a factor tree2 Multiply to last number add
to middle number3 Put out baggies (parenthesis)4 Split first term nice5 Drop in factors from tree
Example
xsup2 + 12x + 32
(x + 8) (x + 4)
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Solve with Graphing Calculator
1) Solve each equation for y (3 easy steps)
2) Use y= button and enter each equation
3) Use graph to eyeball answerOr
4) Use to find where y1 and y2 are equal
5) Be sure answer is in (x y) form
Y=
Systems of EquationsTo Solve by Graphing
1) Make an x y Chart
2) Select any x solve for y x 2x ndash 4 y 0 2(0) ndash 4 -4 1 2(1) ndash 4 -2
3) Then graph the two points4) Do for both equations5) The answer is where they
cross6) Be sure answer is in (x y)
form
2nd TABLE
bull y + 2x = 9
bull y = 9 ndash 2xbull 3(9 - 2x) ndash 2x = 11
bull 27-6x ndash 2x = 11bull 27 ndash 8x = 11bull - 8x = -16bull x = 2bull y = 9 ndash 2 (2)bull y = 5bull (2 5)
Systems of EquationsSolve by Substitution
(box amp shove)
1) Solve one equation for x or y (change sides change signs)
2) Box it3) Rewrite other equation
and shove box in 4) Solve for surviving letter
1) Distribute2) Combine like terms3) Solve
4) Send it back to box5) Solve for other letter6) Answer in (xy) form
3y -2x = 11y + 2x = 9
Systems of EquationsSolve by Elimination
1) Look for opposite signs
2) Multiply to create opposites
3) Add old and new equation together
4) Solve for surviving letter
5) Plug back into either equation (pick the easy one)
6) Solve for other letter
7) Answer in (xy) format
bull 2x ndash y = 93x + 4y = -14
bull 4(2x ndash y = 9)
bull 8x ndash 4y = 36bull 3x + 4y = -14bull 11x = 22bull x = 2
bull 2(2) ndash y = 9bull -y = 5bull y = -5
bull (2 -5)
2x ndash y = 93x + 4y = -14
4 No perfect squares on end 3 terms amp starts with xsup2
NO proceed to question 5
YES its quadratics in the morning
xsup2 - 10x + 24MA
Multiply to +24 Add to -10
38 11
-3 -8 -11
-2 -12 -14
-6-4 -10 Found it
Put in the parenthesis
(x-6) (x-4)
All Done
2 Search and Seizure (quadratics in the morning)(See question 4)
1 Steal the ldquoardquo and give it to the last term (Multiply)1
5 Is there a number in front of the xsup2 amp does it have 3 terms
Yes Jail BreakNO
Then it is Prime
(canrsquot factor)
Example 2xsup2 + 7x + 3
xsup2 + 7x + 6 (23)
( x + 6 ) ( x + 1 )
3 Arrested and Caught - Divide last terms by ldquoardquo
( x + 62 ) ( x + 12 )
4 Beat it Down - (reduce fractions)
( x + 3 ) ( x + 12 )
5 Parole (kick denominator to the front)(x + 3 ) ( 2 x + 1 )
All Done
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Check This First
Is there a common factor (number or letter)
Greatest Common Factor (GCF)
1 Factor each term2 Circle common terms3 Multiply common terms4 Write it down5 Put out baggie for leftover6 Multiply leftovers for each
term7 Put in baggie ( )
Example
8xsup2y + 4xsup3 - 12x
4x ( 2xy + xsup2 - 3 )
Is there a Square on each End
Perfect Squares
1 Put out Parenthesis2 Split FIRST and LAST numbers nice3 Put in Signs
3 Different Kinds
A xsup2 - 16x + 64
(xndash 8) (x ndash 8)
B xsup2 + 18x + 81
(x + 9) (x + 9)
C xsup2 - 36
(x + 6) (x ndash 6)
Is there a number in front of the xsup2 and does it have 3 terms
Jail Break
1Steal the ldquoardquo and give it to the last term (multiply)2Search and Seizure (Quad in AM)3Arrested and Caught (divide by ldquoardquo)4Beat Down (reduce fractions)5Parole (kick denominator to the front)6Check it out (FACE it)
Example
2xsup2 + 7x + 3
1xsup2 + 7x + 62(x + 6) (x + 1)3(x + 62) (x + 12)4(x + 3) (x +12)5(x + 3) (2x + 1)
No perfect squares and 3 terms and starts with xsup2
Quadratics in the Morning (AM)
1 Make a factor tree2 Multiply to last number add
to middle number3 Put out baggies (parenthesis)4 Split first term nice5 Drop in factors from tree
Example
xsup2 + 12x + 32
(x + 8) (x + 4)
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Solve with Graphing Calculator
1) Solve each equation for y (3 easy steps)
2) Use y= button and enter each equation
3) Use graph to eyeball answerOr
4) Use to find where y1 and y2 are equal
5) Be sure answer is in (x y) form
Y=
Systems of EquationsTo Solve by Graphing
1) Make an x y Chart
2) Select any x solve for y x 2x ndash 4 y 0 2(0) ndash 4 -4 1 2(1) ndash 4 -2
3) Then graph the two points4) Do for both equations5) The answer is where they
cross6) Be sure answer is in (x y)
form
2nd TABLE
bull y + 2x = 9
bull y = 9 ndash 2xbull 3(9 - 2x) ndash 2x = 11
bull 27-6x ndash 2x = 11bull 27 ndash 8x = 11bull - 8x = -16bull x = 2bull y = 9 ndash 2 (2)bull y = 5bull (2 5)
Systems of EquationsSolve by Substitution
(box amp shove)
1) Solve one equation for x or y (change sides change signs)
2) Box it3) Rewrite other equation
and shove box in 4) Solve for surviving letter
1) Distribute2) Combine like terms3) Solve
4) Send it back to box5) Solve for other letter6) Answer in (xy) form
3y -2x = 11y + 2x = 9
Systems of EquationsSolve by Elimination
1) Look for opposite signs
2) Multiply to create opposites
3) Add old and new equation together
4) Solve for surviving letter
5) Plug back into either equation (pick the easy one)
6) Solve for other letter
7) Answer in (xy) format
bull 2x ndash y = 93x + 4y = -14
bull 4(2x ndash y = 9)
bull 8x ndash 4y = 36bull 3x + 4y = -14bull 11x = 22bull x = 2
bull 2(2) ndash y = 9bull -y = 5bull y = -5
bull (2 -5)
2x ndash y = 93x + 4y = -14
2 Search and Seizure (quadratics in the morning)(See question 4)
1 Steal the ldquoardquo and give it to the last term (Multiply)1
5 Is there a number in front of the xsup2 amp does it have 3 terms
Yes Jail BreakNO
Then it is Prime
(canrsquot factor)
Example 2xsup2 + 7x + 3
xsup2 + 7x + 6 (23)
( x + 6 ) ( x + 1 )
3 Arrested and Caught - Divide last terms by ldquoardquo
( x + 62 ) ( x + 12 )
4 Beat it Down - (reduce fractions)
( x + 3 ) ( x + 12 )
5 Parole (kick denominator to the front)(x + 3 ) ( 2 x + 1 )
All Done
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Check This First
Is there a common factor (number or letter)
Greatest Common Factor (GCF)
1 Factor each term2 Circle common terms3 Multiply common terms4 Write it down5 Put out baggie for leftover6 Multiply leftovers for each
term7 Put in baggie ( )
Example
8xsup2y + 4xsup3 - 12x
4x ( 2xy + xsup2 - 3 )
Is there a Square on each End
Perfect Squares
1 Put out Parenthesis2 Split FIRST and LAST numbers nice3 Put in Signs
3 Different Kinds
A xsup2 - 16x + 64
(xndash 8) (x ndash 8)
B xsup2 + 18x + 81
(x + 9) (x + 9)
C xsup2 - 36
(x + 6) (x ndash 6)
Is there a number in front of the xsup2 and does it have 3 terms
Jail Break
1Steal the ldquoardquo and give it to the last term (multiply)2Search and Seizure (Quad in AM)3Arrested and Caught (divide by ldquoardquo)4Beat Down (reduce fractions)5Parole (kick denominator to the front)6Check it out (FACE it)
Example
2xsup2 + 7x + 3
1xsup2 + 7x + 62(x + 6) (x + 1)3(x + 62) (x + 12)4(x + 3) (x +12)5(x + 3) (2x + 1)
No perfect squares and 3 terms and starts with xsup2
Quadratics in the Morning (AM)
1 Make a factor tree2 Multiply to last number add
to middle number3 Put out baggies (parenthesis)4 Split first term nice5 Drop in factors from tree
Example
xsup2 + 12x + 32
(x + 8) (x + 4)
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Solve with Graphing Calculator
1) Solve each equation for y (3 easy steps)
2) Use y= button and enter each equation
3) Use graph to eyeball answerOr
4) Use to find where y1 and y2 are equal
5) Be sure answer is in (x y) form
Y=
Systems of EquationsTo Solve by Graphing
1) Make an x y Chart
2) Select any x solve for y x 2x ndash 4 y 0 2(0) ndash 4 -4 1 2(1) ndash 4 -2
3) Then graph the two points4) Do for both equations5) The answer is where they
cross6) Be sure answer is in (x y)
form
2nd TABLE
bull y + 2x = 9
bull y = 9 ndash 2xbull 3(9 - 2x) ndash 2x = 11
bull 27-6x ndash 2x = 11bull 27 ndash 8x = 11bull - 8x = -16bull x = 2bull y = 9 ndash 2 (2)bull y = 5bull (2 5)
Systems of EquationsSolve by Substitution
(box amp shove)
1) Solve one equation for x or y (change sides change signs)
2) Box it3) Rewrite other equation
and shove box in 4) Solve for surviving letter
1) Distribute2) Combine like terms3) Solve
4) Send it back to box5) Solve for other letter6) Answer in (xy) form
3y -2x = 11y + 2x = 9
Systems of EquationsSolve by Elimination
1) Look for opposite signs
2) Multiply to create opposites
3) Add old and new equation together
4) Solve for surviving letter
5) Plug back into either equation (pick the easy one)
6) Solve for other letter
7) Answer in (xy) format
bull 2x ndash y = 93x + 4y = -14
bull 4(2x ndash y = 9)
bull 8x ndash 4y = 36bull 3x + 4y = -14bull 11x = 22bull x = 2
bull 2(2) ndash y = 9bull -y = 5bull y = -5
bull (2 -5)
2x ndash y = 93x + 4y = -14
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Check This First
Is there a common factor (number or letter)
Greatest Common Factor (GCF)
1 Factor each term2 Circle common terms3 Multiply common terms4 Write it down5 Put out baggie for leftover6 Multiply leftovers for each
term7 Put in baggie ( )
Example
8xsup2y + 4xsup3 - 12x
4x ( 2xy + xsup2 - 3 )
Is there a Square on each End
Perfect Squares
1 Put out Parenthesis2 Split FIRST and LAST numbers nice3 Put in Signs
3 Different Kinds
A xsup2 - 16x + 64
(xndash 8) (x ndash 8)
B xsup2 + 18x + 81
(x + 9) (x + 9)
C xsup2 - 36
(x + 6) (x ndash 6)
Is there a number in front of the xsup2 and does it have 3 terms
Jail Break
1Steal the ldquoardquo and give it to the last term (multiply)2Search and Seizure (Quad in AM)3Arrested and Caught (divide by ldquoardquo)4Beat Down (reduce fractions)5Parole (kick denominator to the front)6Check it out (FACE it)
Example
2xsup2 + 7x + 3
1xsup2 + 7x + 62(x + 6) (x + 1)3(x + 62) (x + 12)4(x + 3) (x +12)5(x + 3) (2x + 1)
No perfect squares and 3 terms and starts with xsup2
Quadratics in the Morning (AM)
1 Make a factor tree2 Multiply to last number add
to middle number3 Put out baggies (parenthesis)4 Split first term nice5 Drop in factors from tree
Example
xsup2 + 12x + 32
(x + 8) (x + 4)
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Solve with Graphing Calculator
1) Solve each equation for y (3 easy steps)
2) Use y= button and enter each equation
3) Use graph to eyeball answerOr
4) Use to find where y1 and y2 are equal
5) Be sure answer is in (x y) form
Y=
Systems of EquationsTo Solve by Graphing
1) Make an x y Chart
2) Select any x solve for y x 2x ndash 4 y 0 2(0) ndash 4 -4 1 2(1) ndash 4 -2
3) Then graph the two points4) Do for both equations5) The answer is where they
cross6) Be sure answer is in (x y)
form
2nd TABLE
bull y + 2x = 9
bull y = 9 ndash 2xbull 3(9 - 2x) ndash 2x = 11
bull 27-6x ndash 2x = 11bull 27 ndash 8x = 11bull - 8x = -16bull x = 2bull y = 9 ndash 2 (2)bull y = 5bull (2 5)
Systems of EquationsSolve by Substitution
(box amp shove)
1) Solve one equation for x or y (change sides change signs)
2) Box it3) Rewrite other equation
and shove box in 4) Solve for surviving letter
1) Distribute2) Combine like terms3) Solve
4) Send it back to box5) Solve for other letter6) Answer in (xy) form
3y -2x = 11y + 2x = 9
Systems of EquationsSolve by Elimination
1) Look for opposite signs
2) Multiply to create opposites
3) Add old and new equation together
4) Solve for surviving letter
5) Plug back into either equation (pick the easy one)
6) Solve for other letter
7) Answer in (xy) format
bull 2x ndash y = 93x + 4y = -14
bull 4(2x ndash y = 9)
bull 8x ndash 4y = 36bull 3x + 4y = -14bull 11x = 22bull x = 2
bull 2(2) ndash y = 9bull -y = 5bull y = -5
bull (2 -5)
2x ndash y = 93x + 4y = -14
Is there a number in front of the xsup2 and does it have 3 terms
Jail Break
1Steal the ldquoardquo and give it to the last term (multiply)2Search and Seizure (Quad in AM)3Arrested and Caught (divide by ldquoardquo)4Beat Down (reduce fractions)5Parole (kick denominator to the front)6Check it out (FACE it)
Example
2xsup2 + 7x + 3
1xsup2 + 7x + 62(x + 6) (x + 1)3(x + 62) (x + 12)4(x + 3) (x +12)5(x + 3) (2x + 1)
No perfect squares and 3 terms and starts with xsup2
Quadratics in the Morning (AM)
1 Make a factor tree2 Multiply to last number add
to middle number3 Put out baggies (parenthesis)4 Split first term nice5 Drop in factors from tree
Example
xsup2 + 12x + 32
(x + 8) (x + 4)
The Math ldquoFrdquo Word ldquoFactoringrdquo Summary
Solve with Graphing Calculator
1) Solve each equation for y (3 easy steps)
2) Use y= button and enter each equation
3) Use graph to eyeball answerOr
4) Use to find where y1 and y2 are equal
5) Be sure answer is in (x y) form
Y=
Systems of EquationsTo Solve by Graphing
1) Make an x y Chart
2) Select any x solve for y x 2x ndash 4 y 0 2(0) ndash 4 -4 1 2(1) ndash 4 -2
3) Then graph the two points4) Do for both equations5) The answer is where they
cross6) Be sure answer is in (x y)
form
2nd TABLE
bull y + 2x = 9
bull y = 9 ndash 2xbull 3(9 - 2x) ndash 2x = 11
bull 27-6x ndash 2x = 11bull 27 ndash 8x = 11bull - 8x = -16bull x = 2bull y = 9 ndash 2 (2)bull y = 5bull (2 5)
Systems of EquationsSolve by Substitution
(box amp shove)
1) Solve one equation for x or y (change sides change signs)
2) Box it3) Rewrite other equation
and shove box in 4) Solve for surviving letter
1) Distribute2) Combine like terms3) Solve
4) Send it back to box5) Solve for other letter6) Answer in (xy) form
3y -2x = 11y + 2x = 9
Systems of EquationsSolve by Elimination
1) Look for opposite signs
2) Multiply to create opposites
3) Add old and new equation together
4) Solve for surviving letter
5) Plug back into either equation (pick the easy one)
6) Solve for other letter
7) Answer in (xy) format
bull 2x ndash y = 93x + 4y = -14
bull 4(2x ndash y = 9)
bull 8x ndash 4y = 36bull 3x + 4y = -14bull 11x = 22bull x = 2
bull 2(2) ndash y = 9bull -y = 5bull y = -5
bull (2 -5)
2x ndash y = 93x + 4y = -14
Solve with Graphing Calculator
1) Solve each equation for y (3 easy steps)
2) Use y= button and enter each equation
3) Use graph to eyeball answerOr
4) Use to find where y1 and y2 are equal
5) Be sure answer is in (x y) form
Y=
Systems of EquationsTo Solve by Graphing
1) Make an x y Chart
2) Select any x solve for y x 2x ndash 4 y 0 2(0) ndash 4 -4 1 2(1) ndash 4 -2
3) Then graph the two points4) Do for both equations5) The answer is where they
cross6) Be sure answer is in (x y)
form
2nd TABLE
bull y + 2x = 9
bull y = 9 ndash 2xbull 3(9 - 2x) ndash 2x = 11
bull 27-6x ndash 2x = 11bull 27 ndash 8x = 11bull - 8x = -16bull x = 2bull y = 9 ndash 2 (2)bull y = 5bull (2 5)
Systems of EquationsSolve by Substitution
(box amp shove)
1) Solve one equation for x or y (change sides change signs)
2) Box it3) Rewrite other equation
and shove box in 4) Solve for surviving letter
1) Distribute2) Combine like terms3) Solve
4) Send it back to box5) Solve for other letter6) Answer in (xy) form
3y -2x = 11y + 2x = 9
Systems of EquationsSolve by Elimination
1) Look for opposite signs
2) Multiply to create opposites
3) Add old and new equation together
4) Solve for surviving letter
5) Plug back into either equation (pick the easy one)
6) Solve for other letter
7) Answer in (xy) format
bull 2x ndash y = 93x + 4y = -14
bull 4(2x ndash y = 9)
bull 8x ndash 4y = 36bull 3x + 4y = -14bull 11x = 22bull x = 2
bull 2(2) ndash y = 9bull -y = 5bull y = -5
bull (2 -5)
2x ndash y = 93x + 4y = -14
bull y + 2x = 9
bull y = 9 ndash 2xbull 3(9 - 2x) ndash 2x = 11
bull 27-6x ndash 2x = 11bull 27 ndash 8x = 11bull - 8x = -16bull x = 2bull y = 9 ndash 2 (2)bull y = 5bull (2 5)
Systems of EquationsSolve by Substitution
(box amp shove)
1) Solve one equation for x or y (change sides change signs)
2) Box it3) Rewrite other equation
and shove box in 4) Solve for surviving letter
1) Distribute2) Combine like terms3) Solve
4) Send it back to box5) Solve for other letter6) Answer in (xy) form
3y -2x = 11y + 2x = 9
Systems of EquationsSolve by Elimination
1) Look for opposite signs
2) Multiply to create opposites
3) Add old and new equation together
4) Solve for surviving letter
5) Plug back into either equation (pick the easy one)
6) Solve for other letter
7) Answer in (xy) format
bull 2x ndash y = 93x + 4y = -14
bull 4(2x ndash y = 9)
bull 8x ndash 4y = 36bull 3x + 4y = -14bull 11x = 22bull x = 2
bull 2(2) ndash y = 9bull -y = 5bull y = -5
bull (2 -5)
2x ndash y = 93x + 4y = -14
Systems of EquationsSolve by Elimination
1) Look for opposite signs
2) Multiply to create opposites
3) Add old and new equation together
4) Solve for surviving letter
5) Plug back into either equation (pick the easy one)
6) Solve for other letter
7) Answer in (xy) format
bull 2x ndash y = 93x + 4y = -14
bull 4(2x ndash y = 9)
bull 8x ndash 4y = 36bull 3x + 4y = -14bull 11x = 22bull x = 2
bull 2(2) ndash y = 9bull -y = 5bull y = -5
bull (2 -5)
2x ndash y = 93x + 4y = -14
