Lesson 9, Topic 3
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Equations with Fractions Involving Binomial Denominator

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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

$$\frac{6x}{6} = \frac{-2}{6}$$

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

$$\scriptsize y = 3$$

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

$$\scriptsize 5x \: – \:4x = \: – \: 14 \: + \: 25$$

$$\scriptsize x = \: + \: 25\: – \: 14$$

$$\scriptsize x = 11$$

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