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Equations with fractions

Linear equations

Japanese school year: Junior high 1

What you learn

You will learn how to eliminate denominators by multiplying both sides by a common multiple to solve equations with integer coefficients. This approach is frequently used to handle equations arising from rates, ratios, and percentages. Familiarity with finding least common multiples and properties of equality is helpful.

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Key points

This technique clears fractions from an equation. Multiplying both sides by a common multiple of denominators aa and bb converts coefficients into integers for easier solving.

xa+xb=c  ⟺  bx+ax=abc(ab≠0)\frac{x}{a} + \frac{x}{b} = c \iff bx + ax = abc \quad (ab \neq 0)

This identity cautions that a minus sign in front of a fraction applies to the entire numerator. Placing parentheses around the numerator as (ax+b)(ax + b) prevents sign errors.

−ax+bc=−(ax+b)c-\frac{ax + b}{c} = -\frac{(ax + b)}{c}

This is a key reminder when clearing denominators. To keep the equation balanced, multiply every term on both sides—including plain numbers without fractions—by the common multiple.

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