Equations with Fractions Involving Binomial Denominator

Worked Example 9.3.1:

Solution

i. \( \frac{1}{x\: +\: 3} \:+\: \frac{5}{1}\scriptsize = 0 \)

Find the L.C.M

\( \frac{1(1) \: + \:5(x \: + \:3)}{x \:+ \: 3} \scriptsize = 0 \)

\( \frac{1 \:+ \:5x \: + \:15}{x \: + \: 3} \scriptsize = 0 \)

Cross multiply

\( \scriptsize 1 \; + \; 5x \; + \; 15 = 0 \)

Take like terms

\( \scriptsize 5x \: + \: 16 \: = 0 \)

\( \scriptsize 5x = -16 \)

Divide both sides by 5

\( \frac{5x}{5} = \frac{-16}{5} \)

\( \frac{\not{5}x}{\not{5}} = \frac{-16}{5} \)

\( \scriptsize x = \normalsize \frac{-16}{5} \)

⇒ \( \scriptsize x = \scriptsize -3\frac{1}{5} \)

ii. \( \frac{10}{x \: – \: 2} \scriptsize = 4 \)

\( \scriptsize 10 = 4(x \: – \: 2)\)

\( \scriptsize 10 = 4x \: – \: 8\)

Take like terms

\( \scriptsize 10 \: + \: 8 \: = \: 4x \)

\( \scriptsize 18 = 4x \)

\( \scriptsize 4x =18 \)

Divide both sides by 4 

\( \frac{4x}{4} = \frac{18}{4} \)

\( \frac{\not{4}x}{\not{4}} = \frac{18}{4} \)

\( \frac{\not{4}x}{\not{4}} = \frac{9}{2} \)

⇒ \( \scriptsize x = 4 \frac{1}{2} \)

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

Cross multiply

\( \scriptsize 5 (x\: + \: 2) = 3(2x \: – \: 7) \)

\( \scriptsize 5x \: + \: 10 = 6x \: – \: 21 \)

Take like terms

\( \scriptsize 5x \: – \: 6x = \: – \: 21 \: – \: 10 \)

\( \scriptsize -x \: = \: -31\)

Divide through by -1

\( \frac{-x}{-1} = \frac{-31}{-1} \)

⇒ \( \scriptsize x = 31 \)

iv. \( \frac{1}{3y \: – \: 7} = \frac{2}{4y \: – \: 8} \)

Cross multiply

\( \scriptsize 1(4y \: – \: 8) = 2(3y \: – \: 7) \)

\( \scriptsize 4y \: – \: 8 = 6y \: – \: 14 \)

Take like terms

\( \scriptsize 4y \: – \: 6y = \: – \: 14 \: + \: 8 \)

\( \scriptsize \: – \: 2y = \: – \: 6 \)

Divide both sides by  -2 

\( \frac{-2y}{-2} = \frac{-6}{-2} \)

⇒ \( \scriptsize y = 3 \)

v. \( \frac{2}{1\: + \: 2x} = \frac{4}{1 \: + \: x} \)

Cross multiply

\( \scriptsize 2(1 \: + \: x) = 4(1 \: + \: 2x) \)

\(\scriptsize 2 \: + \: 2x = 4 \: + \: 8x \)

Take like terms

\(\scriptsize 2 \: – \: 4 = 8x \: – \: 2x\)

\(\scriptsize \: – \: 2 = 6x\)

\(\scriptsize 6x = \: – \: 2\)

Divide both sides by 6

\( \frac{6x}{6} = \frac{-2}{6} \)

\( \frac{\not{6}x}{\not{6}} = \frac{-\not{2}^1}{\not{6}^3} \)

\(\scriptsize x = \: – \: \normalsize \frac{1}{3}\)

⇒ \( \scriptsize y = 3 \)

vi. \( \frac{x \: – \: 5}{2x \: – \: 7} = \frac{2}{5} \)

Cross multiply

\( \scriptsize 5(x \: – \: 5) = 2(2x \: – \: 7) \)

\( \scriptsize 5x \: – \: 25 = 4x \: – \: 14 \)

Take like terms

\( \scriptsize 5x \: – \:4x = \: – \: 14 \: + \: 25 \)

\( \scriptsize x = \: + \: 25\: – \: 14 \)

⇒ \( \scriptsize x = 11\)