Solving Equations

When fractions occur we can sometimes transform the equation to one that does not involve fractions.

Example

Find the solution to the equation

(4 x/ 5) - (7 / 4) = ( x/ 5) + ( x/ 4) .

Solution

The least common multiple of the denominators in the equation is 4 × 5 = 20 and we proceed as follows:

16 x - 35 = 4 x + 5 x

16 x - 35 = 9 x

adding 35 to both sides and subtracting 9 x from both sides leads to

7 x = 35 so x = 5 is the solution to the equation .

Exercise

Find the solution to each of the following equations.

(a) 5 x - 6( x - 5) = 2( x + 5) + 5( x - 4)

(b) ( x + 15)( x - 3) - ( x - 6 x + 9) = 30 - 15( x - 1)

(c) ( x - 2) / 2 + ( x + 10) / 9 = 5

Solution

(a)

5 x - 6( x - 5) = 2( x + 5) + 5( x - 4)

5 x - 6 x + 30 = 2 x + 10 + 5 x - 20

- x + 30 = 7 x - 10

30 = x + 7 x - 10

30 = 8 x - 10

30 + 10 = 8 x

8 x = 40

x = 5

(b) First, using FOIL , we expand

( x + 15)( x - 3) = x - 3 x + 15 x - 45 = x + 12 x - 45

Now we have

( x + 15)( x - 3) - ( x14 2 - 6 x + 9) = 30 - 15( x - 1)

x + 12 x - 45 - x + 6 x - 9 = 30 - 15 x + 15

18 x - 54 = 45 - 15 x

18 x + 15 x - 54 = 45

33 x - 54 = 45

33 x = 45 + 54

33 x = 99

x = 3

(c) This time we multiply both sides by 2 × 9 .

9( x - 2) + 2( x + 10) = 90

9 x - 18 + 2 x + 20 = 90

11 x + 2 = 90

11 x = 88

x = 8

Quiz

Which of the following is the solution to the equation (x - 4)/7 = (x - 10)/5 ?

(a) 11 (b) - 10 (c) 19 (d) 25

Solution:

The highest common factor of the denominators is 5 × 7 = 35 . Multiplying both sides of the equation by this

5( x - 4) = 7( x - 10)

5 x - 20 = 7 x - 70

5 x - 20 + 70 = 7 x - 70 + 70

5 x + 50 = 7 x

50 = 7 x - 5 x = 2 x

x = 25

so that x = 25 is the solution. This can be checked by putting this value into the original equation and showing that each side will have the value 3 .