CCBC Math 081
Third Edition
Solving Equations Using the Multiplication Property of Equality
Section 5.5
7 pages
5.5 Solving Equations Using the Multiplication Property of Equality
The Multiplication Property of Equality
Another property needed for solving the equations in one variable is called the
Multiplication Property of Equality. For any real numbers a, b, and c, where c does not
a b
equal 0, if a  b then a  c  b  c and  . This property states that multiplying (or
c c
dividing) both sides of an equation by the same nonzero number produces an equivalent
equation—an equation that has the same solution. We use this property to transform an
equation into a simpler one. Therefore, multiplying (or dividing) the same nonzero
number on both sides of an equation will not change the equation’s solution.
MULTIPLICATION PROPERTY OF EQUALITY
WORDS
SYMBOLS
Multiplying (or dividing) both sides of an
equation by the same nonzero number
produces an equivalent equation.
If a = b,
then
a b
a  c  b  c and  for c  0
c c
Multiplicative Inverse
Recall the Inverse Property of Multiplication that we studied in a previous section.
INVERSE PROPERTY OF MULTIPLICATION
WORDS
The product of a number and its reciprocal is 1.
The product of a term and its reciprocal is 1.
SYMBOLS
EXAMPLE
1
a 1
a
1
5  1
5
1
2x 
1
2x
Multiplicative Inverse: The multiplicative inverse of a is
1
. Similarly, the multiplicative
a
1
is a. We say that the inverse operation of multiplication is division and the
a
inverse operation of division is multiplication.
1
Note: When
and a are multiplied together the answer is 1.
a
inverse of
397
CCBC Math 081
Third Edition
Example 1:
Solving Equations Using the Multiplication Property of Equality
Section 5.5
7 pages
Solve the equation 8x = 56. Then check the solution.
Let’s try to do this by thinking about what the equation says. ―8 times what number is equal
to 56?‖ The answer is 7.
Now, we will learn how to solve this equation algebraically. To solve the equation means to
find the value of the variable that makes the equation a true statement. To do this, we want to
get the variable on one side of the equation by itself; we call this isolating the variable. On
the left side of the equal sign we have the expression 8x. This means 8  x , where 8 is the
coefficient of the variable x. To isolate x, we perform the inverse operation of multiplication:
division. Divide both sides of the equation by 8, the coefficient of x.
8 x  56
8 x 56

8
8
1x  7
Use the Multiplication Property of Equality.
Divide both sides of the equation by 8.
Simplify.
x7
Typically, we do not write the third step: 1x = 7. However, even when it is unwritten, we
are using the idea of the multiplicative identity: multiplying 1 and a number (1 and x in this
problem) produces x, thereby isolating the variable.
Check: Substitute 7 for x in the original equation.
8 x  56
8(7)  56
56  56
This is a true statement. Therefore, x = 7 is the solution of 8x  56 .
Practice 1:
Solve the equation 9x = 54.
Answer: x = 6
Watch It:
http://youtu.be/PkjHu2Z1v0c
Example 2:
Solve the equation –x = 24. Then check the solution.
The expression -x means 1 x , where -1 is the coefficient of the variable x. To isolate x, we
perform the inverse operation of multiplication: division. Divide both sides of the equation
by -1, the coefficient of x.
 x  24
1x  24
1x 24

-1
-1
x  24
Rewrite as multiplication.
Divide both sides by -1.
Simplify.
398
CCBC Math 081
Third Edition
Solving Equations Using the Multiplication Property of Equality
Section 5.5
7 pages
Check: Substitute -24 for x in the original equation.
 x  24
(-24 )  24
24  24
This is a true statement. Therefore, x = -24 is the solution of  x  24 .
Practice 2:
Solve the equation – y = 16.
Watch It:
http://youtu.be/Ux9heoKgRSI
Example 3:
Solve the equation -4x = -18. Then check the solution.
Answer:
y = -16
The coefficient of the variable x is -4. To isolate x, we perform the inverse operation of
multiplication: division. Divide both sides of the equation by -4.
4 x  18
4 x 18

-4
-4
9
x
2
Check: Substitute
Use the Multiplication Property of Equality.
Divide both sides of the equation by -4.
Simplify.
9
for x in the original equation.
2
4 x  18
9
4    18
2
 4
9
 18
1
2
2
 4
9
 18
1 21
 2 9
 18
1 1
Rewrite the left side as a fraction multiplication
problem.
Divide the numerator of the first fraction and the
denominator of the second fraction by 2.
Simplify.
18  18
This is a true statement. Therefore, x =
9
is the solution of -4x = -18.
2
399
CCBC Math 081
Third Edition
Solving Equations Using the Multiplication Property of Equality
Practice 3:
Solve the equation – 96 = – 4a.
Watch It:
http://youtu.be/3KcNA2XnPA0
Answer:
Section 5.5
7 pages
a = 24
x
 4 . Then check the solution.
2
On the left side of the equal sign, x is divided by 2. To isolate x, we perform the inverse
operation of division: multiplication. Multiply both sides of the equation by 2.
x
4
2
x
Multiply both sides of the equation by 2.
2    2(4)
2
 
Rewrite the left side as a fraction multiplication
2 x
 2(4)
problem.
1 2
1
Divide out a 2 in the numerator and denominator.
2 x
 2(4)
1 21
Example 4:
Solve the equation
x 8
Simplify.
Check: Substitute 8 for x in the original equation.
x
4
2
8
4
2
44
This is a true statement. Therefore, x = 8 is the solution of
Watch It:
a
7.
5
http://youtu.be/RKARnaaM0sw
Example 5:
Solve the equation 0.5x  10 .
Practice 4:
Solve the equation
 10
0.5 x
10

-0.5
-0.5
x  20
0.5 x
x
 4.
2
Answer:
a = 35
Use the Multiplication Property of Equality.
Divide both sides of the equation by – 0.5.
Simplify.
Practice 5:
Solve the equation – 1.2w = 6.
Watch It:
http://youtu.be/vUKjtwjwGIY
400
Answer: w = -5
CCBC Math 081
Third Edition
Solving Equations Using the Multiplication Property of Equality
Section 5.5
7 pages
2
Solve the equation x  10 .
3
2
x  10
3
2 
3  x   3(10)
Multiply both sides of the equation by 3.
3 
3 2
Rewrite the left side as a fraction multiplication
x  30
problem.
1 3
Example 6:
1
3 2
x  30
1 31
Divide out a 3 in the numerator and denominator.
2 x  30
Simplify.
2 x 30

2
2
Divide both side by 2.
Simplify.
x  15
This equation could also be solved using the concept of the reciprocal. Remember that if we
multiply a number by its reciprocal the product (answer) is 1. We will do the multiplication
2 3
and division step at the same time by multiplying by the reciprocal. The reciprocal of
is .
3 2
2
x  10
3
3
32  3
Multiply both sides of the equation of .
 x   (10)
2
23  2
3 2
3 10
Rewrite as a fraction multiplication problem.
 x 
2 3
2 1
1
3 21
3 105

x

1
31
12
12
1x  15
x = 15
Practice 6:
Watch It:
Divide out common factors.
Simplify.
3
Solve the equation x  12 .
4
http://youtu.be/ijEa6DIbzqs
Answer:
Watch All: http://youtu.be/ePR9c_7FN5g
401
x = 16
CCBC Math 081
Third Edition
Solving Equations Using the Multiplication Property of Equality
5.5 Solving Equations using the Multiplication Property Exercises
Solve each using the Multiplication Property of Equality.
1.
3x = 24
12.
2.
16x = 4
13.
3.
77 = 11y
4.
48 = 12y
5.
8x = 0
6.
-x = 10
7.
14.
-15 = -6y
1
x6
3
5
x 8
7
15.
0.4 x  12
16.
0.03x  6
-x = -10
17.
0.02 x  18
8.
-6y = 36
18.
9.
-7y = 1
19.
10.
4x = -30
11.
-11z = -121
20.
402
x
8
6
x
3
1.4
1.5x  3
Section 5.5
7 pages
CCBC Math 081
Third Edition
Solving Equations Using the Multiplication Property of Equality
5.5 Solving Equations using the Multiplication Property Exercises Answers
Solve each using the Multiplication Property of Equality.
1.
3.
x=8
1
x
4
7=y
4.
4=y
5.
x=0
15.
56
5
x  30
6.
x = -10
16.
x  200
7.
x = 10
17.
x = -900
8.
y = -6
18.
x = 48
19.
x = 4.2
20.
x = -2
2.
9.
10.
11.
12.
13.
14.
1
7
15
x
2
y
403
z = 11
5
y
2
x  18
x
Section 5.5
7 pages