Equations with Unknown Variables in the Denominator

Worked Example 9.2.1:

i. \( \frac{12}{x} \; – \; \frac{2}{3x} = \frac{1}{6} \)

Find the L.C.M

\( \frac{3(12) \; – \; 1(2)}{3x} = \frac{1}{6} \)

\( \frac{36 \; – \; 2}{3x} = \frac{1}{6} \)

\( \frac{34}{3x} = \frac{1}{6} \)

Cross multiply

\( \scriptsize 3x = 34 \; \times \; 6 \)

\( \scriptsize 3x = 204 \)

Divide both sides by 3

\( \frac{ 3x}{3} = \frac{204}{3} \)

\( \scriptsize x = \normalsize \frac{204}{3} \)

\( \scriptsize x = 68\)

ii. \( \frac{4}{5x} = \frac{1}{35} \: – \: \frac{3}{7} \)

Find the L.C.M of the right side

\( \frac{4}{5x} = \frac{1 \: – \: 15}{35} \)

\( \frac{4}{5x} = \frac{-14}{35} \)

Cross  multiply

\( \scriptsize 4 \; \times \; 35 = 5x \; \times \; -14 \)

\( \scriptsize 140 = -70x \)

Divide both sides by -70

\( \frac{ 140}{-70} = \frac{-70x}{-70} \)

-2 = x

Move x to the left hand side

∴ x = – 2

iii. \( \frac{2}{5x} \: – \: \frac{3}{2x} = \scriptsize 1 \frac{4}{7} \)

Find the L.C.M of the right hand side

\( \frac{2(2) \: – \: 5(3)}{10x} = \frac{11}{7} \)

\( \frac{4 \: – \: 15}{10x} = \frac{11}{7} \)

\( \frac{-11}{10x} = \frac{11}{7} \)

Cross multiply

\( \scriptsize +7 \: \times \: -11 = 10x \: \times \: 11 \)

\( \scriptsize -77 = 110x \)

Divide both sides by 110

\( \frac{ -77}{110} = \frac{110x}{110} \)

Move x to the left-hand side

x = \( \frac{ -77}{110} \)

iv. \( \frac{2}{5} = \frac{1}{2y} \; – \; \frac{3}{4y} \)

Find the L.C.M of the right side

⇒ \( \frac{2}{5} = \frac{2(1) \; -\; 3}{4y} \)

⇒ \( \frac{2}{5} = \frac{2 \; -\; 3}{4y} \)

⇒ \( \frac{2}{5} = \frac{-1}{4y} \)

Cross multiply

\( \scriptsize 2 \; \times \; 4y = 5 \; \times \; -1 \)

\( \scriptsize 8y = -5 \)

Divide both sides by 8

\( \frac{8y}{8} = \frac{-5}{8} \)

\( \frac{\not{8}y}{\not{8}} = \frac{-5}{8} \)

\( \scriptsize y = \normalsize \frac{-5}{8} \)

v. \( \frac{5}{8} = \frac{10}{x}\)

Cross multiply

\( \frac{5 \: \searrow}{8 \: \nearrow} = \frac{\swarrow \: 10}{ \nwarrow \: x}\)

\( \scriptsize 8 \: \times \: 10 = 5x \)

\( \scriptsize 80 = 5x \)

Divide both sides by 5

\( \frac{80}{5} = \frac{5x}{5} \)

\( \scriptsize x = 16\)

vi. \( \frac{1}{x} = \frac{2}{7}\)

Cross multiply 

1 × 7 = 2 × x

7 = 2x

Divide both sides by  2

\( \frac{7}{2} = \frac{2x}{2} \)

\( \frac{7}{2} = \frac{2x}{2} \)

\( \frac{7}{2} = \frac{\not{2}x}{\not{2}} \)

\(\scriptsize x = \normalsize \frac{7}{2}\)

\(\scriptsize x = 3 \frac{1}{2}\)