Lesson 9, Topic 2
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# Equations with Unknown Variables in the Denominator

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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

i.e $$\frac{2}{5} = \frac{2(1) \; -\; 3}{4y}$$

=$$\frac{2}{5} = \frac{2 \; -\; 3}{4y}$$

= $$\frac{2}{5} = \frac{-1}{4y}$$

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 x 7 = 2 x x

7 = 2x

Divide both sides by  2

$$\frac{7}{2} = \frac{2x}{2}$$

$$\frac{7}{2} = \frac{2x}{2}$$

$$\frac{7}{2} = \frac{\not{2}x}{\not{2}}$$

$$\scriptsize x = \normalsize \frac{7}{2}$$

$$\scriptsize x = 3 \frac{1}{2}$$
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