Lesson 3-4 to 3-5 Study Guide
Lesson 3-4: Solving Equations by Multiplying or Dividing
Solving Equations by Dividing: Just as addition and subtraction are inverse operations,
multiplication and division are inverse operations. To isolate the variable in an equation
involving multiplication, you can apply the Division Property of Equality.
Division Property of Equality: If you divide each side of an equation by the same
nonzero number, the two sides remain equal.
Example 1:
Solve -7x = 42. Check your solution.
-7x = 42
−7𝑥
−7
=
42
−7
Write the equation.
Division Property of Equality
1x = -6
-7 ÷ -7 = 1, 42 ÷ -7 = -6
x = -6
Identity Property; 1x = x
check/ -7(-6) = 42
Check solution in the original problem. It is correct.
Solving Equations by Multiplying: To isolate the variable in an equation in which the
variable is being divided, you can apply the Multiplication Property of Equality.
Multiplication Property of Equality: If you multiply each side of an equation by the same
number, the two sides remain equal.
Example 2:
𝒚
Solve
𝑦
2
𝟐
= −𝟐. Check your solution.
= −2
2 ·
𝑦
2
Write the equation.
= −2 · 2
Multiplication Property of Equality
y=-4
check/
−4
2
= −2
Check your solution. It works.
Example 3:
𝟓
Solve − 𝟔 𝒃 = 𝟏𝟓. Check your solution.
𝟓
− 𝟔 𝑏 = 15
6
5
Write the equation.
6
− 5 · − 6 𝑏 = 15 · − 5
6
Multiply each side of the equation by − 5 which is
5
the reciprocal of − 6
b = - 18
𝟓
check/ − 𝟔 · −18 = 15
Check your solution. It works.
Lesson 3-5: Solving Two Step Equations
Solve Two-Step Equations: A two-step equation contains two operations. To solve twostep equations, use inverse operations to undo each operation in reverse order. First,
undo addition/subtraction. Then, undo multiplication/division.
Example 1
Solve
𝑐
2
𝑐
2
𝒄
Solve a Two-Step Equation
− 𝟏𝟑 = 𝟕. Check your solution.
𝟐
− 13 = 7
Write the equation.
+ (−13) = 7
Rewrite subtraction to “add the opposite”
+13
+13
𝑐
2
Addition Property of Equality
= 21
𝑐
2 · 2 = 21 · 2
Multiplication Property of Equality
c = 42
check/
42
2
− 13 = 7
Always check in the original problem. It works.
7=7
Example 2 Solve a Two-Step Equation
y
Solve 8 +
= 14. Check your solution.
3
y
8+
= 14
Write the equation.
3
-8
= -8
Subtraction Property of Equality
y
=6
Simplify.
3
y
3
=6∙3
Multiplication Property of Equality
3
y = 18
Simplify.
check/ 8 +
18
3
= 14
Check your solution. It works.
Example 3
Equations with Negative Coefficients
Solve 6 = 9 – x. Check your solution.
6=9–x
Write the equation.
6 = 9 – 1x
Identity Property; x = 1x
6 = 9 + (-1x)
Rewrite subtraction to “add the opposite”
6 = 9 + (-1x)
Subtraction Property of Equality
-9 -9______
-3 = -1x
Simplify.
3
 1x
=
Division Property of Equality
1
1
3=x
Simplify.
check/ 6=9-3
Check your solution. It works.
Example 4
Equations with Distributive Property
Solve 3(2x + 8) = 114. Check your solution.
3(2x + 8) = 114
6x + 24 = 114
-24 -24
6x = 90
6𝑥
6
=
90
6
x = 15
check/ 3(2•15 + 8) = 114
Example 5
Write the equation.
Use the Distributive Property
Subtraction Property of Equality
Division Property of Equality
Simplify.
Check your solution. It works.
Combine Like Terms Before Solving
Solve 3w – 7w – 11 = 25. Check your solution.
3w – 7w – 11 = 25
Write the equation.
3w +(-7w) + (-11) = 25
Rewrite subtraction to “add the opposite”
-4w +-11 = 25
Combine like terms.
+11 = +11
Addition Property of Equality
-4w = 36
Simplify.
 4w
36
=
Division Property of Equality
4
4
w = -9
Simplify.
check/3(-9) – 7(-9) – 11 = 25
Check your solution in the original equation.
Writing Equations
Example 2
Translate Sentences into Equations
Translate each sentence into an equation.
Sentence
Equation
a. Three less than twice a number is –12.
2n – 3 = -12
b. Mara has 15 ponytail holders, which is 6 more than 3 times
the number that Janelle has.
15 = 3n + 6
c. The quotient of a number and 4, decreased by 2, is 5.
n
4-2=5
Example 3
Write and Solve an Equation
BASKETBALL Olivia scored twenty-four points in a basketball game. This is six
more than three times the number of points that Madison scored. Find the number
of points that Madison scored.
Let n = the number of points that Madison scored. Then, 3n + 6 = the number of points
that Olivia scored.
24 = 3n + 6
–6=
–6
18 = 3n
18
3
=
Write the equation.
Subtraction Property of Equality
3𝑛
Simplify.
3
6=n
Divide each side by 3.
Madison scored 6 points.
Real-World Example 4
Solve a Two-Step Verbal Problem
Two-Step Verbal Problems: Some real-world situations involve a given amount which
then increases or decreases at a certain rate. Such a situation can be represented by a
two-step equation.
PRINTING A laser printer prints 9 pages per minute. Liza refilled the paper tray after it
had printed 92 pages. In how many more minutes will there be a total of 245 pages
printed?
Let m = the number of minutes. Write and solve an equation. The remaining pages to
print is 9m.
9m + 92 =245
____-92 = -92
Write the equation.
Subtraction Property of Equality
9m = 153
9𝑚
9
=
153
9
Division Property of Equality
m = 17 minutes
check/ 9(17) +92 = 245
Check in the original equation. It works.