How do you clear fractions from a linear equation with rational coefficients?

Conditions

• The equation is linear.
• The coefficients are rational numbers (fractions).
• Each equation in a system is processed separately using its own common denominator.

Reasoning, step by step

1. Identify all the denominators present in the equation.
2. Calculate the least common multiple (LCM) of these denominators to find the common denominator.
3. Multiply every term on both sides of the equation by this common denominator.
4. Perform the arithmetic multiplication in the numerators.
5. Reduce each fraction by dividing the numerator by the denominator to obtain integer coefficients.

Example

For the equation $\frac{1}{2}x - \frac{2}{3}y = \frac{7}{3}$, the denominators are 2 and 3, so the common denominator is 6. Multiplying every term by 6 yields $\frac{6}{2}x - \frac{12}{3}y = \frac{42}{3}$, which simplifies to $3x - 4y = 14$.

Common misconceptions

• Forgetting to multiply the constant term on the right side of the equation by the common denominator.
• Changing the sign of a term accidentally during the multiplication step.
• Assuming the same common denominator must be used for all equations in a system, rather than finding one independently for each equation.

Watch the explanation

Explore next

• How do you determine the correct common denominator to use for an equation with multiple fractions?
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• Why is the elimination method preferred after clearing the fractions in this specific system?