Equations with algebraic fractions

Objective. Solve linear equations involving numerical denominators.

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2. Worked example
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Learn: the key idea

Equations involving algebraic fractions have numerical denominators, such as x/3 + 2 = 5, and are solved by first clearing the fractions through multiplying every term by the lowest common denominator. Once the fractions are cleared, the equation becomes a standard linear equation that can be solved using inverse operations. For example, to solve x/4 + 1 = 3, multiply every term by 4 to get x + 4 = 12, then subtract 4 to find x = 8. When more than one fraction appears with different denominators, find their lowest common multiple before multiplying through, keeping the equation balanced on both sides.

Worked example

Solve the equation with fractions: x/5 + 3 = 7.

1 $Subtract 3 from both sides first: x/5 = 4.$
2 $Multiply both sides by 5 to clear the denominator: x = 4 \times 5.$
3 $Calculate: x = 20.$
4 $Check by substituting back: 20/5 + 3 = 4 + 3 = 7, which matches.$

Answer: x = 20

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Common mistake. Pupils often multiply only the fractional term by the denominator and leave the other terms unchanged, producing an unbalanced equation such as x + 2 = 5 instead of x + 8 = 20. Multiply every single term on both sides by the common denominator.

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