reasoning with equations - welcome to ms. sellars...

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www.njctl.org

2013-08-15

Reasoning with Equations

A.SSE.1, A.CED.1, 2, 3, A.REI.1, 3, A.CED.1, 4, F.BF.1

http://www.njctl.org



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Table of Contents

Inverse Operations

One Step Equations

Two Step Equations

Multi-Step Equations

Click on a topic to go to that section.

Distributing Fractions In Equations

Equations with Variables on Both SidesLiteral Equations

Vocabulary

Tables and Expressions

Evaluating Expressions

Translating Between Words and Expressions

Combining Like Terms

The Distributive Property

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Vocabulary

Return to Table of Contents

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What is a Constant?A constant is a fixed value, a number on its own, whose value does not change. A constant may either be positive or negative.

Example: 4x + 2

In this expression 2 is a constant.click to reveal

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What is a Variable?

A variable is any letter or symbol that represents a changeable or unknown value.

Example: 4x + 2

In this expression x is a variable.

click to reveal

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What is a Coefficient?

A coefficient is the number multiplied by the variable. It is located in front of the variable.

Example: 4x + 2

In this expression 4 is a coefficient.click to reveal

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If a variable contains no visible coefficient, the coefficient is 1.

Example 1: x + 7 is the same as 1x + 7

- x + 7 is the same as

-1x + 7

Example 2:

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1 In 2x - 12, the variable is "x"

True

False

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2 In 6y + 20, the variable is "y"

True

False

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3 In 3x + 4, the coefficient is 3

True

False

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4 What is the constant in 7x - 3?

A 7B xC 3D - 3

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5 What is the coefficient in - x + 3?

A noneB 1C -1D 3

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6 x has a coefficient

True

False

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What is an Algebraic Expression?

An Algebraic Expression contains numbers, variables and at least one operation.

Example:

4x + 2 is an algebraic expression.

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What is an Equation?

Example:

4x + 2 = 14

An equation is two expressions balanced with an equal sign.

Expression 1 Expression 2

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An expressions contains:· numbers· variables · operations

Click to reveal

What is the difference between an expression and an equation?

An equation contains:· numbers· variables· operations· an equal sign

Click to reveal

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Translating Between Words and Expressions


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PULL

List words that indicate

addition

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PULL


subtraction

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PULL


multiplication

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PULL


division

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Be aware of the difference between "less" and "less than".

For example:"Eight less three" and "Three less than Eight" are equivalent

expressions. So what is the difference in wording?

Eight less three: 8 - 3 Three less than eight: 8 - 3

When you see "less than", you need to switch the order of the numbers.

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As a rule of thumb, if you see the words "than" or "from" it means you have to reverse the order

of the two items on either side of the word.

Examples: · 8 less than b means b - 8· 3 more than x means x + 3· x less than 2 means 2 - x

click to reveal

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The many ways to represent multiplication...

How do you represent "three times a"?

(3)(a) 3(a) 3 a 3a

The preferred representation is 3a

When a variable is being multiplied by a number, the number (coefficient) is always written in front of the variable.

The following are not allowed:3xa ... The multiplication sign looks like another variablea3 ... The number is always written in front of the variable

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Representation of division...

How do you represent "b divided by 12"?

b ÷ 12

b ∕ 12

b12

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When choosing a variable, there are some that are often avoided:

l, i, t, o, O, s, S

Why might these letters be avoided?

It is best to avoid using letters that might be confused for numbers or operations.

In the case above (1, +, 0, 5)Click

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Three times j

Eight divided by j

j less than 7

5 more than j

4 less than j

12 34 5

6 7

8 9

0+-.÷

Translate the Words into Algebraic Expressions

Using the Red Characters

j

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23 + m

The sum of twenty-three and m

Move

Me

Write the expression for each statement.Then check your answer.

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d - 24

Twenty-four less than d

Move

Me


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4(8-j)

Write the expression for each statement.

**Remember, sometimes you need to use parentheses for a quantity.**

Four times the difference of eight and j

Move

Me

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7w12

The product of seven and w, divided by 12


Move

Me

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(6+p)2


The square of the sum of six and p

Move

Me

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7 The quotient of 200 and the quantity of p times 7

A 200 7p

B 200 - (7p)

C 200 ÷ 7p

D 7p 200

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8 35 multiplied by the quantity r less 45

A 35r - 45

B 35(45) - r

C 35(45 - r)

D 35(r - 45)

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9 Mary had 5 jellybeans for each of 4 friends.

A 5+4 B 5 - 4

C 5 x 4

D 5 ÷ 4

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10 If n + 4 represents an odd integer, the next larger odd integer is represented by

A n + 2B n + 3C n + 5D n + 6

From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.

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11 a less than 27

A 27 - a

B a 27

C a - 27

D 27 + a

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12 If h represents a number, which equation is a correct translation of:“Sixty more than 9 times a number is 375”?

A 9h = 375B 9h + 60 = 375C 9h - 60 = 375D 60h + 9 = 375


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Tables and Expressions


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n

20

40

80

n ÷ 5

4

8

16

Practice Problems!

Complete the table

click

click

click

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n 2n

20

40

100 200

80

40 click

click

click

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n + 11n

10

28

40

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n - 60n

80

120

180

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2a

48

28

24

Mary's age is twice the age of Jack. Use that fact to complete the table.

Jack's Age

12

14

24

a

Mary's Age

Pull Pull

Can you think of an expression containing a variable which determines Mary's age, given Jack's age?

click

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x + 15

$53

$70

$115

The manager of the department store raised the price $15 on each video game.

$100

$38

x

Price after mark up

$55

Original price

Pull Pull

Can you find an expression that will satisfy the total cost of the video game if given the original price?

click

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g - 2

Kindergarten

8th grade

4th grade

A parent wants to figure out the differences in grade level of her two sons. The younger son is two years behind the older one in terms of grade level.

older son's grade level

younger son's grade level

6

10

2

g

Pull Pull

Write an expression containing a variable which

satisfies the difference in grade level of the two boys.

click

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The tire manufacturer must supply four tires for each quad built. Determine the number of quads that can be built, given then number of available tires.

# of tires # of Quads

20

40

100

5

10

25

t t ÷4 or t/4Click

Can you determine an expression containing a variable for the number of quads built based upon the amount of tires available?

Click

Click

Click

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13 Bob has x dollars. Mary has 4 more dollars than Bob. Write an expression for Mary's money.

A 4xB x - 4C x + 4D 4x + 4

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Bob Mary5 912 1627 31

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14 The width of the rectangle is five inches less than its length. The length is x inches. Write an expression for the width.

A 5 - xB x - 5C 5xD x + 5

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15 Frank is 6 inches taller than his younger brother, Pete. Pete's height is P. Write an expression for Frank's height.

A 6PB P + 6C P - 6D 6

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16 The dog weighs three pounds more than twice the cat. Write an expression for the dog's weight. Let c represent the cat's weight.

A 2c + 3B 3c + 2C 2c + 3cD 3c

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17 Write an expression for Mark's test grade. He scored 5 less than Sam. Let x represent Sam's grade.

A 5 - xB x - 5C 5xD 5

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18 Tim ate four more cookies than Alice. Bob ate twice as many cookies as Tim. If x represents the number of cookies Alice ate, which expression represents the number of cookies Bob ate?

A 2 + (x + 4)B 2x + 4C 2(x + 4)D 4(x + 2)


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Evaluating Expressions


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Steps for Evaluating an Expression:

1. Write the expression

2. Substitute the values given for the variables (use parentheses!)

3. Simplify the Expression Remember Order of Operations!

Write - Substitute - Simplify

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3716

Evaluate (4n + 6)2 for n = 1

100

Drag your answer over the green box to check your work. If you are correct, the value will appear.

Write -

Substitute -

Simplify -

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3220

Evaluate the expression 4(n + 6)2 for n = 2

Drag your answer over the green box to check your work. If you are correct, the value will appear.

256

Write -

Substitute -

Simplify -

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108114

130128118

116106

Let x = 8, then use the magic looking glass to reveal the correct value of the expression 12x + 23

104

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118128 130

11420800 72

4x + 2x3

24

Let x = 2, then use the magic looking glass to reveal the correct value of the expression

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19 Evaluate 3h + 2 for h = 3

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20 Evaluate 2(x + 2)2 for x = -10

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21 Evaluate 2x2 for x = 3

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22 Evaluate 4p - 3 for p = 20

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23 Evaluate 3x + 17 when x = -13

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24 Evaluate 3a for a = -12 9

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25 Evaluate 4a + a for a = 8, c = -2 c

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26 If t = -3, then 3t2 + 5t + 6 equals

A -36

B -6

C 6

D 18


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27 What is the value of the expression |−5x + 12| when x = 5?

A -37

B -13

C 13

D 37


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28 What is the value of the expression (a3 + b0)2 when a = −2 and b = 4?A 64B 49C -49D -64


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29 Evaluate 3x + 2y for x = 5 and y = 12

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30 Evaluate 8x + y - 10 for x = and y = 50

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Find the distance using the formula d = r t

Given a rate of 75 mph and a time of 1.5 hours.

1.) Rewrite the expression :

3.) Simplify the expression:

d = r t

d = (75) (1.5)

d = 112.5 miles

2.) Substitute the values for the variables:

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Use the same formula. d = rt, to find the distances using the values below:

r = 62mph t = 2.4 hours




r = 12 mph t = 7.5 hours




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31 Find the distance traveled if the trip took 3 hrs at a rate of 60 mph.

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32 Find the distance traveled if the trip took 1 hr at a rate of 45 mph.

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33 Find the distance traveled if the trip took 1/2 hr at a rate of 50 mph.

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34 Find the distance traveled if the trip took 5 hr at a rate of 50.5 mph.

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35 Find the distance traveled if the trip took 3.5 hr at a rate of 50 mph.

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An Area ModelFind the area of a rectangle whose width is 4 and whose length is x + 2

4

x 2

Area of two rectangles: 4(x) + 4(2) = 4x + 8

4

x + 2

Area of One Rectangle:4(x+2) = 4x + 8

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The Distributive PropertyFinding the area of the rectangles demonstrates the distributive property

4(x + 2) = 4(x) + 4(2) = 4x + 8

The 4 is distributed to each term of the sum (x + 2)

Write an expression equivalent to:

5(x + 3) = 5(x) + 5(3) = 5x + 15

6(x + 4) =

5(x + 7) =

2(x - 1) =

4(x - 8) =

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a(b + c) = ab + ac Example: 2(x + 3) = 2x + 6

(b + c)a = ba + ca Example: (x + 7)3 = 3x + 21

a(b - c) = ab - ac Example: 5(x - 2) = 5x - 10

(b - c)a = ba - ca Example: (x - 3)6 = 6x - 18click to reveal

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The Distributive Property is often used to eliminate the parentheses in expressions like 4(x + 2). This makes it possible to combine like terms in more complicated expressions.

EXAMPLE:3(4x - 6) = 3(4x) - 3(6) = 12 x - 18

-2(x + 3) = -2(x) + -2(3) = -2x + -6 or -2x - 6

-3(4x - 6) = -3(4x) - -3(6) = -12x - -18 or -12x + 18

TRY THESE:4(7x + 5) =

-6(2x + 4) =

-3(5m - 8) =

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Keep in mind that when there is a negative sign out side of the parenthesis it really is a -1.

For example:-(3x + 4) = -1(3x + 4) = -1(3x) + -1(4) = -3x - 4

What do you notice about the original problem and its answer?

The numbers are turned to their opposites.Remove to see answer.

Try these:-(2x + 5) = -(-5x + 3) =

-(6x - 7) = -(-x - 9) =

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36 4(x + 6) = 4 + 4(6)

TrueFalse

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37 Use the distributive property to rewrite the expression without parentheses 2(x + 5)

A 2x + 5B 2x + 10C x + 10D 7x

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38 Use the distributive property to rewrite the expression without parentheses (x - 6)3

A 3x - 6B 3x - 18C x - 18D 15x

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39 Use the distributive property to rewrite the expression without parentheses -4 (x - 9)

A -4x - 36B 4x - 36C -4x + 36D 32x

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40 Use the distributive property to rewrite the expression without parentheses - (4x - 2)

A -4x - 2B 4x - 2C -4x + 2D 4x + 2

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Combining Like Terms


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Like Terms: Terms in an expression that have the same variable raised to the same power

Like Terms

6x and 2x

5y and 8y

4x2 and 7x2

NOT Like Terms

6x and x2

5y and 8

4x2 and x4

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41 Identify all of the terms like 5y

A 5B 4y2

C 18yD 8yE -1y A

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42 Identify all of the terms like 8x

A 5xB 4x2

C 8yD 8E -10x A

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43 Identify all of the terms like 8xy

A 5xB 4x2yC 3xyD 8yE -10xy A

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44 Identify all of the terms like 2y

A 51yB 2wC 3yD 2xE -10y A

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45 Identify all of the terms like 14x 2

A 5xB 2x2

C 3y2

D 2xE -10x2 A

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Simplify by combining like terms

6x + 3x = (6 + 3)x = 9x

5x + 2x = (5 + 2)x = 7x

4 + 5(x + 3) = 4 + 5(x) + 5(3) = 4 + 5x + 15 = 5x + 19

7y - 4y = (7 - 4)y = 3y

Notice that when combining like terms, you add/subtract the coefficients but the variable remains the same.

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Try These:

8x + 9x

7y - 5y

6 + 2x + 12x

7y + 7x

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46 8x + 3x = 11x

TrueFalse

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47 7x + 7y = 14xy

TrueFalse

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48 4x + 4x = 8x2

TrueFalse

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49 -12y + 4y = -8y

TrueFalse

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50 -3 + y + 5 = 2y

TrueFalse

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51 -3y + 5y = 2y

TrueFalse

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52 7x -3(x - 4) = 4x +12

TrueFalse

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53 7 +(x + 2)5 = 5x + 9

TrueFalse

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54 4 +(x - 3)6 = 6x -14

TrueFalse

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55 3x + 2y + 4x + 12 = 9xy + 12

TrueFalse

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56 3x2 + 7x + 5(x + 3) + x2 = 4x2 + 12x + 15

TrueFalse

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57 9x3 + 2x2 + 3(x2 + x) + 5x = 9x3 + 5x2 + 6x

TrueFalse

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58 The lengths of the sides of home plate in a baseball field are represented by the expressions in the accompanying figure.

A 5xyzB x2 + y3zC 2x + 3yzD 2x + 2y + yz

yz

yy

xx


Which expression represents the perimeter of the figure?

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xx+2

x+3

7

xx+2

x+3

7

59 Find the perimeter of the octagon.

A x +24 B 6x + 24 C 24x D 30x

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Inverse Operations


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What is an equation?

An equation is a mathematical statement, in symbols, that two things are exactly the same (or equivalent). Equations are written with an equal sign, as in

2+3=5

9-2=7

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Equations can also be used to state the equality of two expressions containing one or more variables.

In real numbers we can say, for example, that for any given value of x it is true that

4x + 1 = 14 - 1

If x = 3, then

4(3) + 1 = 14 - 1 12 + 1 = 13 13 = 13

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An equation can be compared to a balanced scale.

Both sides need to contain the same quantity in order for it to be "balanced".

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For example, 20 + 30 = 50 represents an equation because both sides simplify to 50. 20 + 30 = 50 50 = 50

Any of the numerical values in the equation can be represented by a variable.

Examples:

20 + c = 50 x + 30 = 50

20 + 30 = y

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Why are we Solving Equations?

First we evaluated expressions where we were given the value of the variable and had to find what the expression simplified to.

Now, we are told what it simplifies to and we need to find the value of the variable.

When solving equations, the goal is to isolate the variable on one side of the equation in order to determine its value (the value that makes the equation true).

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In order to solve an equation containing a variable, you need to use inverse (opposite/undoing) operations on both sides of the equation.

Let's review the inverses of each operation:

Addition Subtraction

Multiplication Division

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To solve for "x" in the following equation... x + 7 = 32

Determine what operation is being shown (in this case, it is addition). Do the inverse to both sides. x + 7 = 32 - 7 -7 x = 25

In the original equation, replace x with 25 and see if it makes the equation true. x + 7 = 32 25 + 7 = 32 32 = 32

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For each equation, write the inverse operation needed to solve for the variable.

a.) y +7 = 14 subtract 7 b.) a - 21 = 10 add 21

c.) 5s = 25 divide by 5 d.) x = 5 multiply by 12 12

move move

move move

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Think about this...

To solve c - 3 = 12

Which method is better? Why?

Kendra

Added 3 to each side of the equation

c - 3 = 12 +3 +3 c = 15

Ted

Subtracted 12 from each side, then added 15.

c - 3 = 12 -12 -12c - 15 = 0 +15 +15 c = 15

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Think about this...

In the expression

To which does the "-" belong?

Does it belong to the x? The 5? Both?

The answer is that there is one negative so it is used once with either the variable or the 5. Generally, we assign it to the 5 to avoid creating a negative variable.

So: Touch to reveal answer

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60 What is the inverse operation needed to solve this equation?

7x = 49

A Addition

B Subtraction

C Multiplication

D Division

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61 What is the inverse operation needed to solve this equation?

x - 3 = -12

A Addition

B Subtraction

C Multiplication

D Division

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One Step Equations


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To solve equations, you must work backwards through the order of operations to find the value of the variable.

Remember to use inverse operations in order to isolate the variable on one side of the equation.

Whatever you do to one side of an equation, you MUST do to the other side!

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Examples:

y + 9 = 16 - 9 -9 The inverse of adding 9 is subtracting 9 y = 7

6m = 72 6 6 The inverse of multiplying by 6 is dividing by 6 m = 12

Remember - whatever you do to one side of an equation, you MUST do to the other!!!

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x - 8 = -2 +8 +8 x = 6

x + 2 = -14 -2 -2 x = -16

2 = x - 6+6 +6 8 = x

7 = x + 3-3 -3 4 = x

15 = x + 17-17 -17 -2 = x

x + 5 = 3 -5 -5 x = -2

One Step EquationsSolve each equation then click the box to see work & solution.

click to showinverse operation






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One Step Equations

3x = 15 3 3 x = 5

-4x = -12 -4 -4 x = 3

-25 = 5x 5 5 -5 = x




x 2x = 20

= 10 (2) (2)

x-6 x = -216

= 36


(-6)(-6)


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62 Solve.

x - 6 = -11

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63 Solve.

j + 15 = -17

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64 Solve.

-115 = -5x

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65 Solve.

= 12 x 9

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66 Solve.

51 = 17y

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67 Solve.

w - 17 = 37

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68 Solve.

-3 = x 7 P

ull

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69 Solve.

23 + t = 11

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70 Solve.

108 = 12r

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Two-Step Equations


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Sometimes it takes more than one step to solve an equation. Remember that to solve equations, you must work backwards through the order of operations to find the value of the variable.

This means that you undo in the opposite order (GEMDAS): 1st: Addition & Subtraction 2nd: Multiplication & Division 3rd: Exponents 4th: Grouping Symbols

Whatever you do to one side of an equation, you MUST do to the other side!

One way to remember the reverse operations is SADMEG

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Examples:

4x + 2 = 10 - 2 - 2 Undo addition first 4x = 8 4 4 Undo multiplication second x = 2

-2y - 9 = -13 + 9 + 9 Undo subtraction first -2y = -4 -2 -2 Undo multiplication second y = 2

Remember - whatever you do to one side of an equation, you MUST do to the other!!!

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5b + 3 = 18 -3 -3 5b = 15 5 5 b = 3

3j - 4 = 14 +4 +4 3j = 18 3 3 j = 6

-2x + 3 = -1 - 3 -3 -2x = -4 -2 -2 x = 2

Two Step EquationsSolve each equation then click the box to see work & solution.

-2m - 4 = 22 +4 +4 -2m = 26 -2 -2 m = -13

+5 = +5

m = 15

w + 6 = 102 -6 -6

w 2 = 4 2 2 w = 8

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71 Solve the equation.

5x - 6 = -56

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14 = 3c + 2

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x 5

- 4 = 24

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5r - 2 = -12

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14 = -2n - 6

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+ 7 = 13 x 5

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+ 2 = -10 x 3

-

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82 Solve

-3 5

1 2

x + = 1 10

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Multi-Step EquationsReturn to Table of Contents

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Steps for Solving Multiple Step EquationsAs equations become more complex, you should:

1. Simplify each side of the equation. (Combining like terms and the distributive property)

2. Use inverse operations to solve the equation.

Remember, whatever you do to one side of an equation, you MUST do to the other side!

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Examples:

5x + 7x + 4 = 28 12x + 4 = 28 Combine Like Terms -4 - 4 Undo Addition 12x = 24 12 12 Undo Multiplication x = 2

-1 = 2r - 7r +19 -1 = -5r + 19 Combine Like Terms-19 = - 19 Undo Subtraction-20 = -5r -5 -5 Undo Multiplication 4 = r

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Try these.

12h - 10h + 7 = 25

-17q + 7q -13 = 27

17 - 9f + 6 = 140

h = 9

q = - 4

f = -13

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Always check to see that both sides of the equation are simplified before you begin solving the equation.

Sometimes, you need to use the distributive property in order to simplify part of the equation.

Remember: The distributive property is a(b + c) = ab + ac

Examples

5(20 + 6) = 5(20) + 5(6) 9(30 - 2) = 9(30) - 9(2)

3(5 + 2x) = 3(5) + 3(2x)

-2(4x - 7) = -2(4x) - (-2)(7)

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Examples:

2(b - 8) = 28 2b - 16 = 28 Distribute the 2 through (b - 8) +16 +16 Undo subtraction 2b = 44 2 2 Undo multiplication b = 22

3r + 4(r - 2) = 13 3r + 4r - 8 = 13 Distribute the 4 through (r - 2) 7r - 8 = 13 Combine Like Terms +8 +8 Undo subtraction 7r = 21 7 7 Undo multiplication r = 3

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Try these.

3(w - 2) = 9

4(2d + 5) = 92

6m + 2(2m + 7) = 54

w = 5

d = 9

m = 4

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85 Solve.

9 + 3x + x = 25

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86 Solve

-8e + 7 +3e = -13

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87 Solve.

-27 = 8x - 4 - 2x - 11

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88 Solve.

n - 2 + 4n - 5 = 13

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89 Solve.

32 = f - 3f + 6f

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90 Solve.6g - 15g + 8 - 19 = -38

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91 Solve.

3(a - 5) = -21

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92 Solve.

4(x + 3) = 20

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93 Solve.

3 = 7(k - 2) + 17

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94 Solve.

2(p + 7) -7 = 5

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95 Solve.

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3m -1m + 3(m-2) = 19.75

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96 Solve.

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97 Solve.

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98 Solve.

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99 Two angles are complementary. One angle is 5 times the other angle. What is the measure in degrees of the larger angle?

Ans

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Distributing Fractions in Equations


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Remember...

1. Simplify each side of the equation.

2. Solve the equation.(Undo addition and subtraction first, multiplication and division second)


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There is more than one way to solve an equation with a fraction coefficient. While you can, you don't need to distribute.

Multiply by the reciprocal Multiply by the LCD

(-3 + 3x) = 3 5

72 5

(-3 + 3x) = 3 5

72 5

(-3 + 3x) = 3 5

72 5

5 3

5 3

-3 + 3x = 24+3 +3 3x = 27 3 3 x = 9

(-3 + 3x) = 3 5

72 5

(-3 + 3x) = 3 5

72 5

5 5

3(-3 + 3x) = 72 -9 + 9x = 72 +9 +9 9x = 81 9 9 x = 9

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Some problems work better when you multiply by the reciprocal and some work better multiplying by the LCM.

Which strategy would you use for the following? Why?

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100 Solve.

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101 Solve.

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(8 - 3c) = 2 3

16 3

102 Solve.

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103 Solve.

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104 Solve.

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Variables on Both Sides


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Remember...

1. Simplify both sides of the equation.

2. Collect the variable terms on one side of the equation. (Add or subtract one of the terms from both sides of the equation)

3. Solve the equation.


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Example:

4x + 8 = 2x + 26-2x -2x Subtract 2x from both sides2x + 8 = 26 - 8 -8 Undo Addition 2x = 18 2 2 Undo Multiplication x = 9

What if you did it a little differently?4x + 8 = 2x + 26-4x -4x Subtract 4x from both sides 8 = -2x + 26 -26 - 26 Undo Addition -18 = -2x -2 -2 Undo Multiplication 9 = x

Recommendation: Cancel the smaller amount of the variable!

Slide down to reveal steps


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Example:

6r - 5 = 7r + 7 - 2r 6r - 5 = 5r + 7 Simplify Each Side of Equation-5r -5r Subtract 5r from both sides (smaller than 6r) r - 5 = 7 + 5 +5 Undo Subtraction r = 12


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Try these:

6x - 2 = x + 13 4(x + 1) = 2x -2 5t - 8 = 9t - 10-x -x 4x + 4 = 2x -2 -5t -5t 5x - 2 = 13 -2x -2x -8 = 4t - 10 + 2 +2 2x + 4 = -2 +10 +105x = 15 -4 -4 2 = 4t 5 5 2x = -6 4 4 x = 3 2 2 = t x = -3

1 2

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Sometimes, you get an interesting answer.What do you think about this?What is the value of x?

3x - 1 = 3x + 1

Since the equation is false, there is " no solution"!

No value will make this equation true.move this

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Sometimes, you get an interesting answer.What do you think about this?What is the value of x?

3(x - 1) = 3x - 3

Since the equation is true, there are infinitely many solutions! The equation is called an identity.

Any value will make this equation true.

move this

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Try these:

4y = 2(y + 1) + 3(y - 1) 14 - (2x + 5) = -2x + 9 9m - 8 = 9m + 4 4y = 2y + 2 + 3y - 3 14 - 2x - 5 = -2x + 9 - 9m - 9m 4y = 5y - 1 9 - 2x = -2x + 9 -8 = 4-5y -5y +2x +2x No Solution -y = -1 9 = 9 y = 1 Identity

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Mary's distance (rate time) equals Jocelyn's distance

(rate time)

Mary and Jocelyn left school at 3:00 p.m. and bicycled home along the same bike path. Mary went at a speed of 12 mph and Jocelyn bicycled at 9 mph. Mary got home 15 minutes before Jocelyn. How long did it take Mary to get home?

Define t = Mary's time in hourst + 0.25 = Jocelyn's time in hours

Relate

Write 12t = 9(t+0.25)

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12t = 9(t + 0.25)

12t = 9t + 2.25-9t -9t

3t = 2.253 3

t = 0.75

It took Mary 0.75h, or 45 min, to get home.

Step 1 - distribute the 9 inside the parenthesis(pull)

Step 2 - subtract 9t from both sides(pull)

Step 3 - divide both sides by 3(pull)

Slide down to reveal work

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105 Solve.

7f + 7 = 3f + 39

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106 Solve.

h - 4 = -5h + 26

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107 Solve.

w - 2 + 3w = 6 + 5w

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108 Solve.

5(x - 5) = 5x + 19

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109 Solve.

-4m + 8 - 2(m + 3) = 4m - 8

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110 Solve.

28 - 7r = 7(4 - r)

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-In the accompanying diagram, the perimeter of ∆MNO is equal to the perimeter of square ABCD. If the sides of the triangle are represented by 4x + 4, 5x - 3, and 17, and one side of the square is represented by 3x, find the length of a side of the square.

5x – 3

4x + 4

17 N

O

M

3x

A B

C D


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Literal Equations


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Formulas show relationships between two or more variables.

A literal equation is an equation in which known quantities are expressed either wholly or in part by means of letters.

Formulas are a prime example of literal equations.

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When solving a literal equation you will be asked to solve for a particular aspect of that equation.

For example, with the formula: I = prt

you might be asked to "solve for p."

This means that p will be on one side of the equation by itself. The new formula will look this:

p = You can transform a formula to describe one quantity in terms of the others by following the same steps as solving an equation.

I rt

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Example:

Transform the formula d = r t to find a formula for time in terms of distance and rate.

What does "time in terms of distance and rate " mean?

d = r tr r

= t d r

Divide both sides by rSlide to reveal steps

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Examples:V = l wh Solve for w P = 2l + 2w Solve for l

V = w P = 2l + 2wl h -2w -2w P - 2w = 2l 2 2 P - 2w = l 2

Slide to revealsteps Slide to

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Example:

To convert Fahrenheit temperature to Celsius, you use the formula:

C = (F - 32)

Transform this formula to find Fahrenheit temperature in terms of Celsius temperature. (see next page)

5 9

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C = (F - 32)

C = F -

+ +

C + = F

C + 32 = F

5 9 5 9

160 9

160 9

160 9

5 9

160 9

9 5

9 5 ( )

9 5

Solve the formula for F

Slide toreveal steps

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Transform the formula for area of a circle to find radius when given Area.

A = r2

= r2

A = r

A Slide to revealanswer

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Solve the equation for the given variable.

m p n q

m p n q

mq p n

= for p

= (q)

=

(q)

2(t + r) = 5 for t

2(t + r) = 5 2 2

t + r =

- r - r

t = - r

5 2

5 2

Move torevealsteps

Move torevealsteps

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111 The formula I = prt gives the amount of simple interest, I, earned by the principal, p, at an annual interest rate, r, over t years.

Solve this formula for p.

A p =

B p =

C p =

D p =

I rt

Irt

Ir t

It r

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112 A satellite's speed as it orbits the Earth is found using the formula . In this formula, m stands for the

mass of the Earth. Transform this formula to find the mass of the Earth.

A m =

B m =

C m =

D m = rv2

G

v2 = Gm r

v2 - r G

rv2 - Gv2

G - r

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113 Solve for t in terms of s

4(t - s) = 7

A t = + s

B t = 28 + s

C t = - s

D t =

7 4

7 47 + s 4

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114 Solve for w

A = lw

A w = Al

B w =

C w =

A l l A

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115 Solve for h

A

B

C

D

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116 Which equation is equivalent to 3x + 4y = 15?

A y = 15 − 3x

B y = 3x − 15

C y = 15 – 3x 4

D y = 3x – 15 4


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117

If , b ≠ 0, then x is equal to

A

B

C

D

From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011

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reasoning with equations - welcome to ms. sellars...

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