1Cademy - Solving $$\frac{3}{4}x = 12$$ by Multiplying by the Reciprocal

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Multiplication Property of Equality

Example

Solving $\frac{3}{4}x = 12$ by Multiplying by the Reciprocal

To solve the equation $\frac{3}{4}x = 12$ , the variable $x$ has the fractional coefficient $\frac{3}{4}$ , so the strategy is to multiply both sides by the reciprocal of $\frac{3}{4}$ , which is $\frac{4}{3}$ .

Step 1 — Multiply both sides by the reciprocal of the coefficient: Apply the Multiplication Property of Equality using $\frac{4}{3}$ :

$\frac{4}{3} \cdot \frac{3}{4}x = \frac{4}{3} \cdot 12$

Step 2 — Simplify using the inverse property of multiplication: On the left side, $\frac{4}{3} \cdot \frac{3}{4} = 1$ , so $1 \cdot x = x$ . On the right side, $\frac{4}{3} \cdot 12 = \frac{48}{3} = 16$ :

$x = 16$

Step 3 — Check by substitution: Replace $x$ with $16$ in the original equation:

$\frac{3}{4}(16) \stackrel{?}{=} 12$

$12 = 12 \checkmark$

Because both sides are equal, $x = 16$ is confirmed as the solution. Note that dividing both sides of $\frac{3}{4}x = 12$ by $\frac{3}{4}$ would also isolate $x$ , but most people find multiplying by the reciprocal to be the easier approach. This technique works because a number multiplied by its reciprocal always equals $1$ , which leaves the variable by itself.

Updated 2026-04-21

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OpenStax Elementary Algebra

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