MM150 Unit 3 Seminar MM150 Unit 3 Seminar

Sections 3.1 - 3.4Sections 3.1 - 3.4

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2

3.13.1

Order of Order of OperationsOperations

2

DefinitionsDefinitions• Algebra: a generalized form of arithmetic.

• Variables: letters used to represent numbers

• Constant: a number on it’s own or a symbol/letter that represents a fixed quantity.

• Algebraic expression: a collection of variables, numbers, parentheses, and/or operation symbols (+ or x).

Expressions DO NOT have equal signs, “=“.

Examples:

• Algebraic equation: is an algebraic expression that has an equal sign, “=“.

Examples: 2 + x = 11 3y – 9 = 36

24 2, 4, 4(3 5), , 8 2

3 5

xx x y y y

x

3

4

Evaluating ExpressionsEvaluating Expressions• Exponents: x2 34 -7y3 59

2 • 2 • 2 • 2 • 2 • 2 • 2, you can rewrite this as 27

x • x • x • x = x4

(2a)(2a)(2a) = (2a)3

(x + 6)(x + 6) = (x + 6)2

• x^2 is the same as x2 2^3 = 23 = 2*2*2 = 8

Be careful!

(-2)4 = (-2)(-2)(-2)(-2) = 16 -24 = -(2*2*2*2) = -16

(-2)4 is not equal to -24

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5

Order of OperationsOrder of Operations1. Parentheses: Perform all operations inside

parentheses or other grouping symbols (use the rules below).

2. Exponents: perform all exponential operations or find any roots.

3. Multiplication/Division: perform all multiplication or division whichever comes first from left to right.

4. Addition/Subtraction: perform all addition or subtraction whichever comes first from left to right.

Please Excuse My Dear Aunt Sally PEMDAS

5

Perform the one that comes first from left to right

Perform the one that comes first from left to

right

Example of EvaluatingExample of Evaluatingan Expressionan Expression

Evaluate the expression x2 + 4x + 5when x = 3.

• Solution:x2 + 4x + 5

= 32 + 4(3) + 5

= 9 + 12 + 5

= 26

Be sure to follow the Order of

Operations!

6

Example of SubstitutingExample of Substitutingfor Two Variablesfor Two Variables

Evaluate when x = 3 and y = 4.

• Solution:

2 24 3 5x xy y

2 2

2 2

4(3) 3(3)(4) 5(4 )

4(9) 36 5(16)

36 36 80

0 80

8

4 3 5

0

x xy y

Be sure to follow the Order of

Operations!

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8

Examples of Checking SolutionsExamples of Checking SolutionsA. Determine if 9 is the solution to 2 + x = 11.

We can check by substituting 9 for x.

2 + x = 11

2 + 9 = 11

11 = 11

B. Determine if 10 a solution to 3y - 9 = 36.

We can check by substituting 10 for y.

3y - 9 = 363(10) - 9 = 3630 - 9 = 3621 =/= 36

8

This is a true statement, therefore 9 is a solution to 2 + x = 11

This is a false statement, therefore 10

is NOT a solution to 3y – 9 = 36

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EVERYONE: page 111 #42EVERYONE: page 111 #42

Cost of a Tour: The cost, in dollars, for Crescent City Tours to provide a tour for x people can be determined by the expression 220 + 2.75x. Determine the cost for Crescent City Tours to provide a tour for 75 people.

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Cost = 220 + 2.75x Substitute

Cost = 220 + 2.75(75) Multiply

Cost = 220 + 206.25 Add

Cost = 426.25 $426.25

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3.23.2

Linear Equations Linear Equations in One Variablein One Variable

10

TermsTerms• Terms: parts that are added or subtracted in an

algebraic expression

• Terms can be:

Constants: 3, - 5, 0, ⅜,

Variables: a, b, c, x, y, z

Products: 3x, ab2, - 99ay5

• Expressions can have:

one term (monomial): x 5t - 10y

two terms (binomial): y + 9 - 6s - 11

three terms (trinomial): x2 + 7x - 10

four terms or more (polynomial): x2y + xy - 11y + 23

NOTE: Decreasing

power of the variable.

Coefficient: numerical part of the term.

Example: in the term 3x, 3 is the coefficient

Example: in - 99ay5, - 99 is the coefficient

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Like and Unlike TermsLike and Unlike Terms• Like Terms: have the same variables with same

exponents on the variables.

5x and 3x are like terms 6ab and -9ab are like terms 16x2 and x2 are like terms -0.35ac5 and -400ac5 are like terms

• Unlike Terms: have different variables or different exponents on the variables.

5x and 3 are unlike terms 6b and -9c are unlike terms 16x and x2 are unlike terms -0.35a5c and -400ac5 are unlike terms

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Example: Combine Like TermsExample: Combine Like Terms

• 8x + 4x= (8 + 4)x= 12x

• 5y - 6y= (5 - 6)y= -y

• x + 15 - 5x + 9= (1- 5)x + (15 + 9)= -4x + 24

• 3x + 2 + 6y - 4 + 7x = (3 + 7)x + 6y + (2 - 4)= 10x + 6y - 2

Add or subtract the coefficients of like terms and KEEP the same variable part

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EVERYONE: page 113 #32EVERYONE: page 113 #32

Combine like terms: 6(r - 3) - 2(r + 5) + 10

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6(r - 3) - 2(r + 5) + 10 Distribute

= 6r - 18 - 2r - 10 + 10 Combine like

terms

= 4r – 18 Finished!

Note that 4r and - 18 are unlike terms, therefore you cannot combine them

15

Addition Property of EqualityAddition Property of Equality

For real numbers a, b, and cif a = b, then a + c = b + c.

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• Example: Solve x - 9 = 15

x - 9 = 15 x - 9 + 9 = 15 + 9 addition

property

x + 0 = 24 x = 24

1616

Subtraction Property of EqualitySubtraction Property of Equality

For real numbers a, b, and cif a = b, then a - c = b – c.

• Example: Solve x + 11 = 19

x + 11 = 19 x + 11 - 11 = 19 - 11 subtraction property

x = 8

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Multiplication Property of Multiplication Property of EqualityEquality

For real numbers a, b, and c, where c =/= 0if a = b, then a • c = b • c

Example: Solve

Multiplication Property

17

9.7

x

x

79

7x

7

7(9)

1 7 x1 7

63

x 63

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Division Property of Division Property of EqualityEquality

For real numbers a, b, and c, where c =/= 0

if a = b, then

Example: Solve 4x = 48

18

a b

c c

4

4 48

4 48

124

x

x

x

Division Property

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Steps for Solving Steps for Solving EquationsEquations

1. Simplify (clean up) both sides of the equation by:a.) Get rid of any fractions by multiplying both sides of the

equation by the LCD.b.) Use the distributive property to get rid of parentheses

when necessary.c.) Combine like terms on same side of equal sign when

possible.Equation will be in the form ax + b = cx + d

2. Collect all the variables on one side of the equal sign and all constants to the other side by using the addition/subtraction property.

Equation will be in the form ax = b

3. Solve for the variable using the division/multiplication property.

The resulting form will be x = c19

EVERYONE: Solve for xEVERYONE: Solve for x 3 3xx - 4 = 17 - 4 = 17

4

3 4 17

3 4 17

3 21

3 21

4

3 37

x

x

x

x

x

20

21 6 3( 2)

21 6 3 6

21 3 12

1221 3 12

9 3

9 3

3

12

3 3

x

x

x

x

x

x

x

EVERYONE: Solve for xEVERYONE: Solve for x 21 = 6 + 3( 21 = 6 + 3(xx + 2) + 2)

21

8 3 6 21

8 3 6 21

8 6 18

8 6 18

2 18

2 18

9

3 3

6 6

2 2

x x

x x

x x

x x

x

x

x

x

x

EVERYONE: Solve for xEVERYONE: Solve for x 8 8xx + 3 = 6 + 3 = 6xx + 21 + 21

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ProportionsProportions

• A proportion is a statement of equality between two ratios.

• Cross Multiplication

If then ad = bc,

b =/= 0, d =/= 0.

,a c

b d

a

b

c

d

a

b

c

d

b • c a • d

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To Solve Application ProblemsTo Solve Application ProblemsUsing ProportionsUsing Proportions

1. Represent the unknown quantity by a variable.

2. Set up the proportion by listing the given ratio on the left-hand side of the equal sign and the unknown and other given quantity on the right-hand side of the equal sign.

When setting up the proportion, the same respective quantities should occupy the

same respective positions on the left and right.

For example, an acceptable proportion might be miles miles

hour hour

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To Solve Application ProblemsTo Solve Application ProblemsUsing Proportions (continued)Using Proportions (continued)

3. Once the proportion is properly written, drop the units and use cross multiplication to solve the equation.

4. Answer the question or questions asked using appropriate units.

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ExampleExample• A 50 pound bag of fertilizer will cover an

area of 15,000 ft2. How many pounds are needed to cover an area of 226,000 ft2?

754 pounds of fertilizer would

be needed.

2 2

50 pounds

15,000 ft 226,000 ft

(50)(226,000) 15,000

11,300,000 15,000

11,300,000 15,000

15,000 15,000

753.33

x

x

x

x

x

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27

3.33.3

FormulasFormulas27

PerimeterPerimeter• The formula for the perimeter of a rectangle is

Perimeter = 2(length) + 2(width) or P = 2L + 2W

• Use the formula to find the perimeter of a yard when L = 150 feet and W = 100 feet.

P = 2L + 2WP = 2(150) + 2(100)P = 300 + 200P = 500 feet

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Volume of a CylinderVolume of a Cylinder• The formula for the volume of a cylinder is

V = (pi)(r2)(h).

• Use the formula to find the height of a cylinder with a radius of 6 inches and a volume of 565.49 in3.

The height of the cylinder is 5 inches.

2

2565.49 (6 )

565.49 36

565.49 36

36 365.000

V r h

h

h

h

h

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30

-9x + 4y = 11

9x - 9x + 4y = 9x + 11

4y = 9x + 11

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EVERYONE: Solve the equation for EVERYONE: Solve the equation for yy

-9x + 4y = 11-9x + 4y = 11

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5x + 3y - 2z = 22

-5x + 5x + 3y - 2z = -5x + 22

3y - 2z = -5x + 22

3y - 2z + 2z = -5x + 22 + 2z

3y = -5x + 2z + 22

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EVERYONE: Solve the equation for EVERYONE: Solve the equation for yy

5x + 3y - 2z = 225x + 3y - 2z = 22

9 9

3

3 8 9 0

3 8 9 0

3 8 9

3

8 8

3 8 9

8 9 3

8 9 3

9 3

8

x x

x y

x y

x y

x y

y x

y x

xy

EVERYONE: Solve the equation for EVERYONE: Solve the equation for yy

33xx + 8 + 8yy - 9 = 0 - 9 = 0

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Solve for Solve for bb2.2.1 2( )2

hA b b

1 2

1 2

1 2

1 2

1 2

1 2

( )2

2 2 ( )2

2 ( )

( )2

2

2

hA b b

hA b b

A h b b

h b bA

h hA

b bh

Ab b

h

33

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3.43.4

Applications of Applications of Linear Equations in Linear Equations in

One VariableOne Variable34

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Translating to MathTranslating to Math• Six more than a number 6 + x

• A number increased by 3 x + 3

• Four less than a number x – 4

• Twice a number 2x

• Four times a number 4x

• 3 decreased by a number 3 – x

• The difference between a number and 5 x – 5

• Four less than 3 times a number 3x – 4

• Ten more than twice a number 2x + 10

• The sum of 5 times a number and 3 5x + 3

• Eight times a number, decreased by 7 8x – 7

• Six more than a number is 10 x + 6 = 10

• Five less than a number is 20 x – 5 = 20

• Twice a number, decreased by 6 is 12 2x – 6 = 12

• A number decreased by 13 is 6 times the number x – 13 = 6x

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To Solve a Word ProblemTo Solve a Word Problem1. Read the problem carefully at least twice to be sure

that you understand it.

2. If possible, draw a sketch to help visualize the problem.

3. Determine which quantity you are being asked to find. Choose a letter to represent this unknown quantity. Write down exactly what this letter represents.

4. Write the word problem as an equation.

5. Solve the equation for the unknown quantity.

6. Answer the question or questions asked.

7. Check the solution.

ExampleExample

The bill (parts and labor) for the repairs of a car was $496.50. The cost of the parts was $339. The cost of the labor was $45 per hour. How many hours were billed?

Let h = the number of hours billedCost of parts + labor = total amount

339 + 45h = 496.5037

Example continuedExample continued

The car was worked on for 3.5 hours.

339 45 496.50

339 339 45 496.50 339

45 157.50

45 157.50

45 453.5

h

h

h

h

h

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ExampleExample

Sandra Cone wants to fence in a rectangular region in her backyard for her lambs. She only has 184 feet of fencing to use for the perimeter of the region.

What should the dimensions of the region be if she wants the length to be 8 feet greater than the width?

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Example continuedExample continued

184 feet of fencing, length 8 feet longer than width

• Let x = width of region• Let x + 8 = length• P = 2L + 2W

x + 8

x

184 2( ) 2( 8)

184 2 2 16

184 4 16

168 4

42

x x

x x

x

x

x

The width of the region is 42 feet and the length is

50 feet.

40

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Page 139 #34Page 139 #34• PetSmart has a sale offering 10% off of all

pet supplies. If Amanda spent $15.72 on pet supplies before tax, what was the price of the pet supplies before the discount?

• the price before discount will be called “x”.

x - x (0.10) = 15.72 x - 0.10x = 15.72

0.9x = 15.72 x ≈17.47 41

The price is≈ $17.47

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Page 140 #46Page 140 #46• A bookcase with three shelves is built by a student. If the height

of the bookcase is to be 2 ft longer than the length of a shelf and the total amount of wood to be used is 32 ft, find the dimensions of the bookcase.

• Let x = width (length of shelf) and let x + 2 = height

• From picture in book, there are 4 pieces of wood for width and 2 pieces of wood for the height.

• 4(width) + 2(height) = total amount of wood

4x + 2(x + 2) = 32

4x + 2x + 4 = 32

6x + 4 = 32

6x = 28

x =

42

So, width of bookcase is ft

and height is ft