Solving Linear Equations with Fractions Revision Notes for Leaving Cert Mathematics

Solving Linear Equations with Fractions

When solving linear equations with fractions, in general, it is best to reduce to one fractions, and then to reduce to no fractions at all.

Example

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Solve for $x$ :

\frac{2x}{3} - \frac{5}{4} = \frac{x}{6} + 1

First, eliminate the fractions. Find the least common denominator (LCD) for all fractions. The denominators are $3,4,6$ .

\text{LCD}=12

Multiply through by $12$ to eliminate fractions.

12 \cdot \left( \frac{2x}{3} \right) - 12 \cdot \left( \frac{5}{4} \right) = 12 \cdot \left( \frac{x}{6} \right) + 12 \cdot 1

4 \cdot 2x - 3 \cdot 5 = 2 \cdot x + 12

8x - 15 = 2x + 12

Collect like-terms :

8x - 2x = 12 + 15

6x = 27

Solve for $x$ :

x = \frac{27}{6}= \frac{9}{2}

Example

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Solve for $x$ :

\frac{3}{x} + 2 = \frac{5}{x} - 1

Multiply through $x$ to eliminate the fractions :

x \cdot \frac{3}{x} + x \cdot 2 = x \cdot \frac{5}{x} - x \cdot 1

3 + 2x = 5 - x

Collect like-terms :

2x + x = 5 - 3

3x = 2

Solve for $x$ :

x = \frac{2}{3}