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12/8/2011
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Semester Review
Fall 2011
Linear Equations
2
How can you tell a linear equation from other equations?
A linear equation can be written in the forms:
Ax By C+ =where A, B, and C are constants and x and y are variables.
(numbers)
GENERAL FORM:
1.
2. A linear equation graphs a straight line.
y mx b= +SLOPE-INTERCEPT FORM:
where m is the slope and b is the y-intercept.
POINT-SLOPE FORM:1 1( )y y m x x− = −
where m is the slope and (x1,y1) is a point.
Slope of a Line
Slope of a line:
3
rise
run
o
o ( )2 2,x y
( )1 1,x yrun
rise
xx
yym =
−−=
12
12
Note: The slope of a line is the SAME everywhere on the line!!! You may use any two
points on the line to find the slope.
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Calculating Slope
4
Example 1: Calculate the slope between the given points.
(3, 7) and (1, 7)?
Intercepts of a Line
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y-intercept: where the graph crosses the y-
axis.
The coordinates are (0, b).
x-intercept: where the graph crosses the x-
axis.
The coordinates are (a, 0).
Recall that there are three possibilities for the manner in which the graphs of two linear equations could meet. The lines could…
consistent system – a system of equation with a solution
inconsistent system – a system of equation without a solution
dependent equations –equations that are equivalent
independent equations –equations that are not equivalent
intersect once
(we have seen this)
not intersect at all
(be parallel)
intersect an infinite number of times (the two lines are the same)
inconsistent
independent
consistent
dependent
consistent
independent
7.2 + 7.3 Solving Systems of Linear Equations
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4 6 5
8 3 9
x y
x y
+ = − =
Is the system consistent or inconsistent?
Are the equations dependent or independent?
Example: Use the elimination method to solve the system of linear equations.
7.3 Solving Systems of Linear Equations Using the Elimination Method
Solve With Calc. and Check
Systems of Linear Inequalities that is unbound.
Is it Bound/unbound?; Consistent/inconsistent and explain?
Dependent/independent and explain?; What is the solution?
Systems of Linear Inequalities
Systems of Linear Inequalities that is bound.
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Solve by Graphing and Check
Explain reasoning
Example 2 (find the domain)
Example 2: Find the domain of function f defined by
Solution to Example 2x2 – x – 2 = 0
(x – 2)(x + 1) = 0
x = 2 or x = –1
Know how to factor!!!
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Example 3 (find the domain)
The domain is all values that x can take on. I cannot have a negative inside the square root. So I'll set the insides greater-than-or-equal-to zero, and solve. The result will be my domain:
–2x + 3 > 0 –2x > –3 2x < 3 x < 3/2
Then the domain is “all x < 3/2”.
Examples:Simplify each of the following; express all answers so that exponents are positive. Whenever an exponent is 0 or negative, we assume the base is not zero.
34 3
3 7
3r r
r r
−
−
−
33
1
6
3
xy
x y−
4.2 Zero and Negative Integer Exponents
( )322
33
23
6yx
y
xxy =
=
9
3
4
273
rr
r −=
−=
Adding Fractions
� Add the two new numerators. Keep the new denominator.
Do Not Add the Denominators!
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1.3 Graph Real Numbers on the Number Line
Example:Graph each set of numbers on the number line.
{ }18 , -2 , 2.75 , 17 , 0.6666... , -5π − −
1.7 Solving Word Problems Involving Real Numbers
Algebraic expression -
Translating from Verbal Form to Algebraic Form
Addition Subtraction Multiplication Division
sum difference product quotient
plus minus times divided by
added to less than twice, etc. ratio
more than decreased by double, triple, etc. split k equal ways
greater than reduced by of average (add first)
increased by subtracted from increased by a factor of
decreased by a factor ofexceeds less
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An all-day parking meter takes only dimes and quarters. If it contains 100 coins with a total value of $14.50, how many of each type of coin are in the meter?
0.15x = 4.50x =4.50/0.15 = 30Thus, there will be 30 quarters and 70 dimes
Let x = number of quarters in the meter. Then100 – x= number of dimes in the meter.
Now, 0.25x + 0.10(100 – x) = 14.50 or0.25x + 10 – 0.10x = 14.50
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2.2 Change Decimals to Percents and Visa Versa
To solve a word problem involving percents, we must first change the percents to their decimal equivalents.
Example:Give the decimal equivalent of 5%.
Examples:Change each percent to a decimal.
55%
100 5 100
0.05
=
= ÷=
Percent means “per 100”
To change a percent to its decimal equivalent, move the decimal point two places to the left.
35%
3.5%
350%
35 %
2.2 Solve Word Problems that Involve Percents
To solve problems involving percents, we will use the formula:
Examples:Solve each equation.
What number is 60% of 200? What percent of 325 is 143?
amount = percent base⋅ Usually comes after the word “of”
don’t forget to change to a decimal
.6(200) = 120 325x = 143
2.5 Use Formulas to Solve Word Problems
We will use the following formulas from business:
We will use the following formulas from science:
retail price = cost + markup OR r c m= +
profit = revenue - cost OR pr c= −
distance = rate time OR drt⋅ =
5( 32)
9
FC
−=
interest = principal rate time OR i prt⋅ ⋅ =
Where C and F are the temperature in degrees Celsius and Fahrenheit respectively.
simple interest formula
Don’t forget to change your percent to a decimal before
using it in a formula!
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2.5 Use Formulas to Solve Word Problems
Geometry Formulas to Know
Rectangle
Perimeter = sum of all sides
or
Perimeter 2 2l w= +
Area = l w⋅
Rectangular Box
Volume l w h= ⋅ ⋅
in units
in 2units
in 3units
2.5 Use Formulas to Solve Word Problems
Geometry Formulas to Know
Rectangle
Perimeter = sum of all sides
or
Perimeter 2 2l w= +
Area = l w⋅
Rectangular Box
Volume l w h= ⋅ ⋅
in units
in 2units
in 3units
Parallelogram
Perimeter = sum of all sides
2.5 Use Formulas to Solve Word Problems
Geometry Formulas to Know
12Area = bh Area bh=
Triangle
Perimeter = sum of all sides
For these formulas: The height meets the base-side at a 90-degree angle
in units
in 2units
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2.5 Use Formulas to Solve Word Problems
Geometry Formulas to Know
Trapezoid
Perimeter = sum of all sides
a+bArea = h
2⋅
Circle
2Area rπ=
Circumference dπ=
3.14159...π ≈227π ≈
*To minimize rounding-errors we will use the
button on our TIπ
in units
2.5 Use Formulas to Solve Word Problems
Geometry Formulas to Know
Cone21
3Volume r hπ=
Pyramid*1
3
*
Volume
B is the area of the base
hB=
base
2.5 Use Formulas to Solve Word Problems
Geometry Formulas to Know
Cylinder2Volume r hπ=
Sphere34
3Volume rπ=
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Example
� Zack is building a gate. It is to be five feet tall and eight feet wide. If the gate is “square” (that is, if the sides meet at the corners to form right angles), what will be the length of the diagonal bracing wire? Round to the nearest quarter-inch.
52 + 82 = c2
25 + 64 = 89 = c2
A collection of 33 coins, consisting of nickels, dimes, and quarters, has a value
of $3.30. If there are three times as many nickels as quarters, and one-half as
many dimes as nickels, how many coins of each kind are there?
Step 1: number of quarters: q
number of nickels: 3q
number of dimes: (½)(3q) = (3/2)q
There is a total of 33 coins, so:
q + 3q + (3/2)q = 33
4q + (3/2)q = 33
8q + 3q = 66
11q = 66
q = 6
Then there are six quarters, and I can work backwards to figure
out that there are 9 dimes and 18 nickels.
How many ounces of pure water must be added to 50 ounces of a 15%
saline solution to make a saline solution that is 10% salt?
ounces
liquid
% salt total ounces salt
water x 0 0
15% sol'n 50 0.15 (50)(0.15) = 7.5
10% mix 50 + x 0.10 0.10(50 + x)
From the last column, you get the equation 7.5 = 0.1(50 + x). Solve for x.
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Helpful Hint for FactoringWhen factoring the sum or difference of two cubes, the sign between
the terms in the binomial factor will be the same as the sign between
the terms.
The sign of the ab term will be the opposite of the sign between the
terms of the binomial factor.
a3 + b3 = (a + b) (a2 – ab + b2)
same sign
The last term in the trinomial will always be positive.
a3 – b3 = (a – b) (a2 + ab + b2)
same sign
opposite sign always positive
opposite sign always positive
Example
� X2 + 7x + 12
� (x + )(x + )
Factors of 12 Sum of factors
1, 12 13
2, 6 8
3, 4 7
Factoring Trinomials
� x2 + x + 3x +3
� Factored form (x + 3)(x + 1) = x2 + 4x + 3
� Notice the product of 3 and 1 = 3 and the sum of 3 and 1 is 4
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Product Rule for Radicals
Examples:
3333 424832 ==
4444 3231648 ==
3333 252125250 −=−=−
and numbers real enonnegativFor nnn abba
ba
=⋅
,
Product Rule for Radicals
Examples:
3233 633 63 222816 xyyxyx ==
4 324 28164 3118 21632 yxyxyx −=−
4 3274 22 yxyx−=
Quotient Rule for Radicals
Examples:
4
2
3 12
3 6
312
6 46464
y
x
y
x
y
x ==
241
216
1
32
2
64
2
64 22
3
5
3
5
xxx
x
x
x
x ====
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The Distance Formula
� d2 = (x2– x
1)2 + (y
2– y
1)2
� Example: Find the distance between the points (-4, 5) and (3, -1)
22 )51()]4(3[ −−+−−=d
~ 9.219544457
Rules of Exponents
For all real numbers a and b and all rational numbers m and n,
Zero exponent rule: a0 = 1, a ≠0
Raising a power to a power:
Raising a product to a power :
Raising a quotient to a power :
(((( )))) nmnm aa ⋅⋅⋅⋅====
(((( )))) mmm baab ====
0 , ≠≠≠≠====
b
ba
ba
m
mm
The Product Rule for Exponents
� Multiplying exponents with the same base:
x3 · x4 = x · x · x x · x · x · x therefore,
= x · x · x · x · x · x · x
= ?
� The product rule for exponents states: to multiply two exponential expressions with the same base, keep the common base and add the exponents.
xm · xn = xm+n
x7
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The Quotient Rule for Exponents
� State that when you divide like bases you subtract their exponents.
� If m and n represent natural numbers, m>n, and x ≠ 0, then
NOTE: you always take the numerator’s exponent minus your denominator’s exponent, NOT the other way around.
nmn
m
xx
x −=
The Power Rule for Exponents
� States when you raise a base to two exponents, you multiply those exponents together.
� If m and n represent natural numbers, then
� Example:
( ) mnnm xx =
( )24z x8
Laws of Exponents: For any integersm, n (assuming no divisions by 0)
m nx x =m
n
x
x=
( )nmx =
Laws of Exponents examples
( )nxy =
nx
y
=
Add
Subtract
Multiply
Multiply
Multiply
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Polynomial Degrees
� Second-degree polynomial, 4x2, x2 – 9, or
ax2 + bx + c
� Third-degree polynomial, –6x3 or x3 – 27
� Fourth-degree polynomial, x4 or 2x4 – 3x2 + 9
� Fifth-degree polynomial, 2x5 or x5 – 4x3 – x + 7
Monomial, Binomial, and Trinomial
Type Definition Example
Monomial A polynomial with one term 5x
Binomial A polynomial with two terms 5x – 10
Trinomial A polynomial with three terms
Examples
32)( 3 +−+−= xxxxh
Find:
a) h(0)
b) h(-3)
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Special Products
� Square of the sums:
(x + y)2 = x2 + 2xy +y2
� The square of the differences:
(x – y )2 = x2 – 2xy +y2
� Product of the sum and difference of two terms:
(a + b)(a – b) = a2 – b2
Identifying Polynomials
The degree of a term of a polynomial in one
variable is the exponent on the variable in that
term.
Example:
5x6 (Sixth) 4x3 (Third) 7x (First) 9 (Zero)
The degree of a polynomial is the same as that of
its highest-degree term.
Example:
5x6 + 4x3 – 7x + 9 (Sixth)
Dividing Polynomials
To divide a polynomial by a polynomial, use the
same method as when performing long division.
40-t-6t52t
52t40-t-6t
Divide.
2
2
+=+
dividend
divisor
1. Divide 6t 2 by 2t. Write the quotient above the term containing the t.
3t
2. Multiply the 3t by 2t + 5. Write the product under the like terms.
6t2 + 15t
3. Subtract. Bring down the remaining term.
-16t - 40
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Dividing Polynomials
40-t-6t52t
52t40-t-6t
Divide.
2
2
+=+
3t
6t2 +15t
-16t - 40
0
83t52t40-t-6t 2
−=+
Check your answer using the FOIL method.
-16t - 40 4. Repeat, using the first term in the bottom row:
- 8
-16t ÷ 2t = - 8
"Percent of" Word Problems
� What percent of 20 is 30?
� What is 35% of 80?
� 45% of what is 9?
30 = (x)(20)
30 ÷ 20 = x = 1.5
So Thirty is 150% of 20
x = (0.35)(80)
x = 28
Twenty-eight is 35% of 80.
9 = (0.45)(x)
9 ÷ 0.45 = x = 20
Nine is 45% of 20.
"Percent of" Word Problems
� A computer software retailer used a markup rate
of 40%. Find the selling price of a computer game
that cost the retailer $25.
The markup is 40% of the $25 cost, so the markup is:
(0.40)(25) = 10
Then the selling price, being the cost plus markup, is:
25 + 10 = 35
The item sold for $35.
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Word Problem: Age
� In three more years, Jon's grandfather will be six times as old as Jon was last year. When Jon's present age is added to his grandfather's present age, the total is 68. How old is each one now?
� One-half of Heather's age two years from now plus one-third of her age three years ago is twenty years. How old is she now?
Word Problem: Area
� A square has an area of sixteen square centimeters. What is the length of each of its sides?
Word Problem: Coin
� A collection of 33 coins, consisting of nickels, dimes, and quarters, has a value of $3.30. If there are three times as many nickels as quarters, and one-half as many dimes as nickels, how many coins of each kind are there?
� A wallet contains the same number of pennies, nickels, and dimes. The coins total $1.44. How many of each type of coin does the wallet contain?
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Word Problem: Distance
� A 555-mile, 5-hour plane trip was flown at two speeds. For the first part of the trip, the average speed was 105 mph. Then the tailwind picked up, and the remainder of the trip was flown at an average speed of 115 mph. For how long did the plane fly at each speed?
Word Problem: Distance
� Two cyclists start at the same time from opposite ends of a course that is 45 miles long. One cyclist is riding at 14 mph and the second cyclist is riding at 16 mph. How long after they begin will they meet?
Word Problem: Mixture
� How many liters of a 70% alcohol solution must be added to 50 liters of a 40% alcohol solution to produce a 50% alcohol solution?
� How many ounces of pure water must be added to 50 ounces of a 15% saline solution to make a saline solution that is 10% salt?
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Word Problem: Numbers
� The product of two consecutive negative even integers is 24. Find the numbers.
� The sum of two consecutive integers is 15. Find the numbers.
Word Problem: Investment
� You put $1000 into an investment yielding 6% annual interest; you left the money in for two years. How much interest do you get at the end of those two years?
� You have $50,000 to invest, and two funds that you’d like to invest in. The You-Risk-It Fund (Fund Y) yields 14% interest. The Extra-Dull Fund (Fund X) yields 6% interest. Because of tax implications, you don't think you can afford to earn more than $4,500 in interest income this year. How much should you put in each fund?
Solving Linear Equations
� Solve 7x + 2 = –54
� Solve 5 + 4x – 7 = 4x – 2 –x
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Graph
� Graph 4x – 3y = 12
� Graph x = 4
� Graph y = 0
Slope-Intercept Form
� Find the equation of the straight line that has slope m= 4 and passes through the point (–1, –6).
Slope-Intercept Form
� Find the equation of the line that passes through the points (–2, 4) and (1, 2).
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Solving Linear Inequalities
� Solve x + 3 < 0, give interval notation, and graph
Solving Linear Inequalities
� Solve 10 <3x + 4 <19
Things to remember
1. Show work and check
2. Interval notation (x, y)
3. Use ruler, bring extra batteries
4. Pencils
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Preparing for the test
� Review previous homework
� Review class notes
� Review concepts and definitions
� Complete the Chapter Review
� Place yourself in test-like conditions
� Get a good night’s sleep
� Allow plenty of time to arrive for test
Taking the test
� Read directions carefully
� Read each problem carefully
� Watch your time
� If you have time, check work and answers
� Do not turn your test in early
� If possible, double-check your work
