Equations with Fractions

To solve equations involving fractions, the first step is to clear the fractions by multiplying every term on both sides of the equation by the LCM of the denominators.

Example 1.3.1:

Solve the following equations:

(i) \(\scriptsize 2y \: + \: 3 \: – \: \normalsize \frac {y}{3} \scriptsize = y \:+ \:7 \)

(ii) \( \frac{x \: – \: 3}{3} \: – \: \frac{x \: – \: 5}{5} \scriptsize = 2 \)

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

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

(WAEC)

Solution:

(i) \(\scriptsize 2y \: + \: 3 \: – \: \normalsize \frac {y}{3} \scriptsize = y \:+ \:7 \)

Clear the fraction: multiply both sides by 3

⇒ \(\scriptsize 2y \: \times \: 3 \: + \: 3 \: \times \: 3\; – \: \normalsize \frac {y}{3} \scriptsize \: \times \: 3 \scriptsize = y \: \times \: 3 \:+ \:7 \: \times \: 3 \)

i.e. 6y + 9 – y = 3y + 21

Collect like terms

⇒ 6y – y – 3y = 21 – 9

⇒ 6y – 4y = 12

⇒ 2y = 12

Divide both sides by 2

⇒ \( \frac {2y}{2} = \frac {12}{2} \)

⇒ y = 6

(ii) \( \frac{x \: – \: 3}{3} \: – \: \frac{x \: – \: 5}{5} \scriptsize = 2 \)

Clear fractions by multiplying both sides by the L.C.M 15

i.e. \( \frac{x \: – \: 3}{3} \scriptsize \: \times \: 15 \; – \: \normalsize \frac{x \: – \: 5}{5}\scriptsize \: \times \: 15 \scriptsize = 2 \: \times \: 15 \)

⇒ 5(x – 3) – 3(x – 5) = 30

Expand the brackets

⇒ 5x – 15 – 3x + 15 = 30

Collect like terms

⇒ 5x – 3x = 30 +15 -15

⇒ 2x = 30

Divide both sides by 2

⇒ \( \frac {2x}{2} = \frac {30}{2} \)

x = 15

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

Clear fractions by multiplying both sides by the LCM 20

i.e. \(\scriptsize 20 \: \times \: \normalsize \frac{2}{5} \scriptsize (x \:+ \:5) = 20 \: \times \: \normalsize \frac{1}{4} \scriptsize (5x \: – \:3) \)

⇒ 8(x + 5) = 5(5x – 3)

Expand the brackets

⇒ 8x + 40 = 25x – 15

Collect like terms

i.e. 40 + 15 = 25x – 8x

⇒ 55 = 17x

⇒ 17x = 55

Divide both sides by 17

⇒ \( \frac {17x}{17} = \frac {55}{7} \)

= \( \scriptsize 3 \normalsize \frac{4}{17} \)

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

Clear fraction: multiply both sides by 12 (L.C.M)

i.e \(\scriptsize 12 \: \times \: \normalsize \frac{2y \: – \: 1}{3} \scriptsize \: – \:12 \; \times \: \normalsize \frac{3 \: – \: y}{2} \scriptsize = \normalsize \frac{y}{4}\scriptsize \: \times \: 12 \)

⇒ 4(2y – 1) – 6(3 – y) = 3y

Expand the brackets

⇒ 8y – 4 – 18 + 6y = 3y

Collect like terms

⇒ 8y + 6y – 3y = 18 + 4

⇒ 11y = 22

Divide both sides by 11

⇒ \( \frac {11y}{11} = \frac {22}{11} \)

∴ y = 2