Basic Mathematics for College Students

8.3 Solving Equations Using Properties of Equality 663

25

 3

75

Since the product of a number and its reciprocal is 1, we can solve equations such
as 23 x 6 , where the coefficient of the variable term is a fraction, as follows.

EXAMPLE 6

Solve: a. b.

StrategyWe will use a property of equality to isolate the variable on one side of
the equation.

WHYTo solve the original equation, we want to find a simpler equivalent
equation of the form , whose solution is obvious.

Solution

a.Since the coefficient of is , we can isolate by multiplying both sides of the

equation by the reciprocal of , which is.

This is the equation to solve.

To undo the multiplication by , multiply both sides

by the reciprocal of.

Use the associative property of multiplication to group and.

On the left side, the product of a number and its reciprocal is 1:

. On the right side,.
The coefficient 1 need not be written since.

Check: This is the original equation.

Substitute 9 for in the original equation.

On the left side,.

Since the statement 66 is true, 9 is the solution of^23 x6.

6  6 23 (9)^183  6

x

2

3

( 9 ) 6

2

3

x 6

x 9 1 xx

(^3 2)  32  1 32  6  (^182)  9
1 x 9
a 32 23

3

2



2

3

b x

3

2

 6

23

3 23

2



2

3

x

3

2

 6

2

3

x 6

23 32

x 23 x

xa number



5

4

x 3

2

3

x 6

Self Check 6

Solve:

a.

b.

Now TryProblems 61 and 67



3

8

b 2

7

2

x 21

WHYTo solve the original equation, we want to find a simpler equivalent
equation of the form , whose solution is obvious.

Solution
To isolate , we use the multiplication property of equality. We can undo the
division by 3 by multiplying both sides by 3.

This is the equation to solve.

Multiply both sides by 3.

Write 3 as.

Do the multiplications.

Simplify by removing the common factor of 3

in the numerator and denominator:.

The coefficient 1 need not be written since.
If we substitute 75 for in , we obtain the true statement. This
verifies that 75 is the solution.

x x 3  25 25  25

x 75 1 xx

3

1

x

31 x

33 x

1 x 75

3 x

3

 75

31

3

1



x

3

 3  25

3 

x

3

 3  25

x

3

 25

x

xa number