Basic Mathematics for College Students

8.3 Solving Equations Using Properties of Equality 665

Since division by 2 is the same as multiplication by , we can also solve

using the multiplication property of equality. We could also isolate by

multiplying both sides by the reciprocalof 2, which is :

b.To isolate on the right side, we use the division property of equality. We can
undo the multiplication by by dividing both sides by.
This is the equation to solve.

On the left side, do the division. The quotient of

two negative numbers is positive. On the right side,

simplify by removing the common factor of 8.6

from the numerator and denominator:

The solution is 0.7. Verify that this is correct by checking.

8.6

1

t

8.6 1 t

0.7t

Use the division property of equality:

(^) Divide both sides by 8.6.

6.02

8.6



8.6t

8.6

6.028.6t

8.6 8.6

t

1

2

 2 t

1

2

 80

12

2 t 80 t

12

0.7

86 60.2

 60 2

0

 

Success Tip It is usually easier to multiply on each side if the coefficient of

the variable term is a fraction,and divide on each side if the coefficient is an

integeror decimal.

EXAMPLE (^8) Solve:
StrategyThe variable is not isolated, because there is a sign in front of it.
Since the term has an understood coefficient of , the equation can be written
as. We need to select a property of equality and use it to isolate the
variable on one side of the equation.
WHYTo find the solution of the original equation, we want to find a simpler
equivalent equation of the form , whose solution is obvious.
SolutionTo isolate , we can either multiply or divide both sides by.
Multiply both sides by : Divide both sides by :
The equation to solve The equation to solve
Write: Write:
On the left side,.
Check: This is the original equation.
Substitute for.
On the left side, the opposite of is 3.
Since the statement 33 is true, 3 is the solution ofx 3.

3  3  3

( 3 ) 3  3 x

x 3

x 3 x 3

 1

1 x 3 1 x (^3)  1  1
 1 x
 1



3

 1

(1)( 1 x)(1) 3

 1 x 3 x 1 x  1 x 3 x 1 x

x 3 x 3

 1  1

x  1

xa number

 1 x 3

x  1

x 

x 3

1.yes 2. 49 3. a. 33 b. 49 4. a. b. 5. 72 6. a. 6 b.

7. a. 11 b.25.1 8. 12

(^2915) 0.5  (^163)
ANSWERS TO SELF CHECKS
Self Check 8
Solve:
Now TryProblem 81
h 12