1Cademy - Solving $$\frac{1}{6}y - \frac{1}{3} = \frac{5}{6}$$ by Clearing Fractions

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Strategy for Solving Equations with Fraction or Decimal Coefficients

Example

Solving $\frac{1}{6}y - \frac{1}{3} = \frac{5}{6}$ by Clearing Fractions

To solve $\frac{1}{6}y - \frac{1}{3} = \frac{5}{6}$ , apply the clearing fractions technique to remove all fractions before isolating the variable.

Step 1 — Find the LCD: The equation contains fractions with denominators $6$ , $3$ , and $6$ . The least common denominator of $6$ and $3$ is $6$ .

Step 2 — Multiply both sides by $6$ and distribute: Multiply each side of the equation by $6$ :

$6\left(\frac{1}{6}y - \frac{1}{3}\right) = 6 \cdot \frac{5}{6}$

Apply the Distributive Property on the left side so that $6$ multiplies each term individually:

$6 \cdot \frac{1}{6}y - 6 \cdot \frac{1}{3} = 6 \cdot \frac{5}{6}$

Simplify each product: $6 \cdot \frac{1}{6} = 1$ , so the first term becomes $y$ ; $6 \cdot \frac{1}{3} = 2$ ; and $6 \cdot \frac{5}{6} = 5$ . The equation is now fraction-free:

$y - 2 = 5$

Step 3 — Solve the cleared equation: Add $2$ to both sides using the Addition Property of Equality:

$y - 2 + 2 = 5 + 2$

$y = 7$

The solution is $y = 7$ . This example demonstrates the power of the clearing technique: multiplying every term by the LCD of $6$ transformed an equation with three fractions into the simple equation $y - 2 = 5$ , which requires just one additional step to solve.

Updated 2026-04-21

Contributors are:

Claude Opus

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Who are from:

University of Michigan - Ann Arbor

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References

OpenStax Elementary Algebra

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