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Solving Equations

Solving Equations: The Addition Property

Definitions

An equation is a mathematical statement in which two
expression shave the same value .

Example: 3x + 4 = 2x + 10

The solution of an equation is a value of the variable that
makes the equation a true statement .

Is 6 is a solution of 3x + 4 = 2x + 10?

Determining Solutions

To determine whether is 6 is a solution of 3x+4=2x+10,
we substitute 6 for x in the equation and simplify.

Ifbothsidesareequal,6isasolution.

Ifbothsidesarenotequal,6isnotasolution.

 
Substitute 6 for each x
Simplify
True!

Solving Equations

Goal: Get the variable alone on one side of the equation and a
number on the other side so the equation reads:

variable = number

Tools: Addition and Multiplication Properties

We use the Addition Property to move terms from one side
of the equation to the other side.

We use the Multiplication Property to solve for the variable or
rewrite an equation.

The Addition Property of Equality

For all real numbers a, b, and c ,

if a = b, then a + c = b + c

The same number may be added or subtracted on both sides of an
equation with out changing the solution .

To solve an equation using the addition property, add (or
subtract) to get the variable term alone on one side of the
equation and the number on the opposite side.

Numbers on both sides?

Move the number.

Example 1 Solve:−15 + x = 37

Adding 15 on both sides gets "x"
alone on one side.

Variable terms on both sides?

Move the variable term.

Example 2 Solve: 2k = −16 + k

Subtracting k on both sides gets k
alone on one side.

Numbers and variables on both sides?

1) Move the variable term to one side.
2) Move the number to the opposite side.

Example 3 Solve: 4y + 1.8 = −5.4 + 3y

Subtracting 3y on both sides gets
y alone on one side.
Subtracting 1.8 on both sides moves
the number to the other side.

Hint: Moving the smaller variable term over to the larger, keeps
the variable term positive!

Practice Exercise 1

Solve: −7= b + 15

A.22

B.8

C.−8

D.−22

Practice Exercise 2

Solve: 4y= 3y − 5.7

A.−5.7

B.57

C.5.7

D.−57

Practice Exercise 3

Solve: 3z + 2.5 = 1.4 + 2z

A.−3.9

B.−1.1

C.3.9

D.1.1

Simplifying Equations

Combine like terms on the same side. (Don't add opposites!!) Then
use the addition property to move terms.

Example 4 Solve: 6 – 3x = 7x – 8 – 9x

Combine: 7x − 9x = −2x
Add 3x on both sides to move the
variable term.
Add 8 on both sides to move the
number.

Removing Parentheses

Use the distributive property to remove parentheses.

Example 5Solve: 6(x −2) + 5 = 5(x + 3)

Distribute 6 and 5.
Combine like terms .
Subtract 5x on both sides.
Add 7 on both sides.

Practice Exercise 4

Solve: 7x –6 + 2x = 2 + 8x –12

A.4

B.8

C.−8

D.−4

Practice Exercise 5

Solve: 8(x −2) = 7(x −3)

A.5

B.−37

C.−5

D.1

Practice Exercise 6

Solve: −3(m + 5) = 7 − 4(3 + m)

A.10

B.−10

C.20

D.5

Solving Equations with Fractions

Example 6 Solve:

 
Subtract 3/4 from both sides.
Find the common denominator : LCD = 12
Write equivalent fractions .
 

Practice Exercise 7

Solve:

A. −5/13
B. 1/40
C. −31/40
D. −1/3

Practice Exercise 8

Solve:

A. −12
B. −18
C. 18
D. 12

Solution − Practice Exercise 1

Solve: −7= b + 15

Answer: D

Subtracting 15on both sides gets b alone on one side.

Solution − Practice Exercise 2

Solve: 4y= 3y − 5.7

Answer: A

Subtracting 3y on both sides gets y alone on one side.

Solution − Practice Exercise 3

Solve: 3z + 2.5 = 1.4 + 2z

Answer: B

Subtract 2z on both sides.
Subtract 2.5 on both sides.

Solution − Practice Exercise 4

Solve: 7x −6 + 2x = 2 + 8x −12

Answer: D

Combine like terms.
Subtract 8x on both sides.
Add 6 on both sides.

Solution − Practice Exercise 5

Solve: 8(x −2) = 7(x −3)

Answer: C

Distribute 8 and 7.
Subtract 7x on both sides.
Add 16 on both sides.

Solution − Practice Exercise 6

Solve: −3(m + 5) = 7 − 4(3 + m)

Answer:  A

Distribute−3and−4.
Combine7and−12.
Add4monbothsidestomovethevariableterm.
Add15onbothsidestomovethenumber.

Solution − Practice Exercise 7

Solve:

Answer: C

 
Subtract 2/5 from both sides
Find the common denominator.
Multiply the denominators; cross
multiply to get the numerators
Simplify.

Solution − Practice Exercise 8

Solve:

Answer: B

 
 Add on both sides.
Simplify.
Subtract 3on both sides.
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