Linear Equations Containing Fractions

To solve simple equations containing fractions, first eliminate the fractions by multiplying each term with the LCM of the denominators and then solve as usual. 

Example 6.3.1:

Solve the following equation 

a. \( \frac{15}{x} \scriptsize = 3 \)

b. \( \frac{x}{7} \scriptsize = 4 \)

c. \( \frac{3x \: + \: 5}{5} \scriptsize = 4 \)

d. \( \normalsize \frac{2}{3} \scriptsize x \: – \: \normalsize \frac{8}{5} \scriptsize = 5 \)

e. \( \normalsize \frac{3}{2x} \scriptsize \: + \: \normalsize \frac{1}{6x} \scriptsize = \normalsize \frac{2}{3} \)

f. \( \normalsize \frac{2x \: – \: 5}{4} \scriptsize \: – \: \normalsize \frac{6x}{2} \scriptsize = 0 \)

Solution 

a. \( \frac{15}{x} \scriptsize = 3 \)

We have only one denominator, multiply through by the denominator (x)

\( \frac{15}{x} \scriptsize \: \times \: x = 3 \: \times \: x \)

\( \frac{15}{\not{x}} \scriptsize \: \times \: \not{\!x} = 3 \: \times \: x \)

\( \scriptsize 15 = 3x \)

Let’s swap the equation around and move 3x to the left-hand side

∴ \( \scriptsize 3x = 15 \)

Divide both sides by 3 

\( \frac{3x}{3} = \frac{15}{3} \)

\( \frac{\not{3}x}{\not{3}} = \frac{15}{3} \)

\( \scriptsize x = \normalsize \frac{15}{3} \)

\( \scriptsize x = 5 \)

b. \( \frac{x}{7} \scriptsize = 4 \)

Denominator = 7 

Multiply through by 7 

\( \frac{x}{7} \scriptsize \: \times \: 7 = 4 \: \times \: 7\)

\( \frac{x}{\not{7}} \scriptsize \: \times \: \not{\!7} = 4 \: \times \: 7\)

\( \scriptsize x = 4 \: \times \: 7\)

\( \scriptsize x = 28 \)

c. \( \frac{3x \: + \: 5}{5} \scriptsize = 4 \)

Denominator = 5 

Multiply through by 5 

\( \frac{3x \: + \: 5}{5} \scriptsize \: \times \: 5 = 4 \: \times \: 5 \)

\( \frac{3x \: + \: 5}{\not{5}} \scriptsize \: \times \: \not{\!5} = 4 \: \times \: 5 \)

\( \scriptsize 3x \: + \: 5 = 4 \: \times \: 5 \)

\( \scriptsize 3x \: + \: 5 = 20\)

Collect like terms 

\( \scriptsize 3x = 20 \: – \: 5\)

\( \scriptsize 3x = 15\)

Divide both sides by 3 

\( \frac{3x}{3} = \frac{15}{3} \)

\( \frac{\not{3}x}{\not{3}} = \frac{15}{3} \)

\( \scriptsize x = \normalsize\frac{15}{3}\)

\( \scriptsize x = 5\)

d. \( \normalsize \frac{2}{3} \scriptsize x \: – \: \normalsize \frac{8}{5} \scriptsize = 5 \)

Denominators = 3, 5

LCM of 3 and 5 = 15 

Multiply through by 15 

\( \normalsize \frac{2}{3} \scriptsize x \: \times \: 15 \: – \: \normalsize \frac{8}{5} \scriptsize \: \times \: 15 = 5 \: \times \: 15 \)

\( \normalsize \frac{2}{\not{3}} \scriptsize x \: \times \: \not{\!15}^5 \: – \: \normalsize \frac{8}{\not{5}} \scriptsize \: \times \: \not{\!15}^3 = 5 \: \times \: 15 \)

\( \scriptsize 2x \: \times \: 5 \: – \: 8 \: \times \: 3 = 75 \)

\( \scriptsize 10x \: – \: 24 = 75 \)

Collect like terms 

\( \scriptsize 10x = 75 \: + \: 24\)

\( \scriptsize 10x = 99\)

Divide both sides by 10

\( \frac{10x}{10} = \frac{99}{10}\)

\( \scriptsize x = \normalsize \frac{99}{10}\)

\( \scriptsize x = 9 \frac{9}{10}\)

e. \( \normalsize \frac{3}{2x} \scriptsize \: + \: \normalsize \frac{1}{6x} \scriptsize = \normalsize \frac{2}{3} \)

Denominators = 2x, 6x and 3 

LCM of 2x, 6x and 3 = 6x

Multiply through by 6x

\( \normalsize \frac{3}{2x} \scriptsize \: \times \: 6x \: + \: \normalsize \frac{1}{6x} \scriptsize \: \times \: 6x = \normalsize \frac{2}{3} \scriptsize \: \times \: 6x \)

\( \scriptsize 3 \: \times \: 3 \: + \: 1 \: \times \: 1 = 2 \: \times \: 2x \)

\( \scriptsize 9 \: + \: 1 = 4x \)

\( \scriptsize 10 = 4x \)

\( \scriptsize 4x = 10 \)

Divide both sides by 4

\( \frac{4x}{4} = \frac{10}{4} \)

\( \frac{\not{4}x}{\not{4}} = \frac{10}{4} \)

\( \scriptsize x = \normalsize \frac{10}{4} \)

\( \scriptsize x = \normalsize \frac{5}{2} \)

\( \scriptsize x = 2\frac{1}{2} \)

f. \( \normalsize \frac{2x \: – \: 5}{4} \scriptsize \: – \: \normalsize \frac{6x}{2} \scriptsize = 0 \)

Denominators = 4 and 2

LCM of 4 and 2 = 4 

Multiply through by 4

\( \normalsize \frac{2x \: – \: 5}{4} \scriptsize \: \times \: 4 \: – \: \normalsize \frac{6x}{2} \scriptsize \: \times \: 4 = 0 \: \times \: 4 \)

\( \normalsize \frac{2x \: – \: 5}{\not{4}} \scriptsize \: \times \: \not{\!4} \: – \: \normalsize \frac{6x}{\not{2}} \scriptsize \: \times \: \not{\!4}^2 = 0 \: \times \: 4 \)

\( \scriptsize 2x \: – \: 5 \: -\: 6x \: \times \: 2 = 0 \)

\( \scriptsize 2x \: – \: 5 \: -\: 12x = 0 \)

Rearrange

\( \scriptsize 2x \: – \:12x \: -\: 5 = 0 \)

\( \scriptsize -10x \: -\: 5 = 0 \)

Collect like terms 

\( \scriptsize -10x = 0 \: + \: 5 \)

\( \scriptsize -10x = 5 \)

Divide both sides by -10

\( \frac{-10x}{-10} = \frac{5}{-10} \)

\( \scriptsize x = \normalsize \frac{5}{-10} \)

\( \scriptsize x = \: \: – \normalsize \frac{1}{2} \)

Evaluation Questions: