1Cademy - Solving $$5 = \frac{1}{2}y + \frac{2}{3}y - \frac{3}{4}y$$ by Clearing Fractions

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Strategy for Solving Equations with Fraction or Decimal Coefficients

Example

Solving $5 = \frac{1}{2}y + \frac{2}{3}y - \frac{3}{4}y$ by Clearing Fractions

To solve the linear equation $5 = \frac{1}{2}y + \frac{2}{3}y - \frac{3}{4}y$ , use the strategy of clearing fractions first.

Step 1: Find the least common denominator (LCD) of all the fractions in the equation. The denominators are $2$ , $3$ , and $4$ . The LCD is $12$ .

Step 2: Multiply both sides of the equation by $12$ to clear the fractions: $12(5) = 12\left(\frac{1}{2}y + \frac{2}{3}y - \frac{3}{4}y\right)$

Distribute the $12$ to every term on the right side: $12(5) = 12\left(\frac{1}{2}y\right) + 12\left(\frac{2}{3}y\right) - 12\left(\frac{3}{4}y\right)$

Simplify each product. Notice there are no more fractions: $60 = 6y + 8y - 9y$

Step 3: Solve the resulting equation. Combine like terms on the right side: $60 = 5y$

Divide both sides by $5$ to isolate $y$ : $\frac{60}{5} = \frac{5y}{5}$ $12 = y$

Check the solution by substituting $y = 12$ into the original equation: $5 \stackrel{?}{=} \frac{1}{2}(12) + \frac{2}{3}(12) - \frac{3}{4}(12)$ $5 \stackrel{?}{=} 6 + 8 - 9$ $5 = 5 \checkmark$ Because both sides are equal, $y = 12$ is verified as the correct solution.

Updated 2026-05-02

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OpenStax Intermediate Algebra

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