Math · College algebra · Worked example

Solve a linear equation with fractions

Solve (2x − 1)/3 − (x + 2)/4 = x/6, then check the answer.

\frac{2 x - 1}{3} - \frac{x + 2}{4} = \frac{x}{6}

Find the least common denominator

The denominators are 3, 4 and 6. The smallest number all three divide into is 12, so multiplying every term by 12 clears the fractions. Any common multiple would work; the least one keeps the numbers small.

LCD (3, 4, 6) = 12

Multiply every term by 12

Multiply each of the three terms, on both sides. Since 12 ÷ 3 = 4, 12 ÷ 4 = 3 and 12 ÷ 6 = 2, each fraction becomes a whole-number multiple of its numerator. Keep the numerators in parentheses: the minus sign in front of the second fraction applies to both x and 2.

\begin{array}{l} 12 \cdot \frac{2 x - 1}{3} - 12 \cdot \frac{x + 2}{4} = 12 \cdot \frac{x}{6} \\ 4 (2 x - 1) - 3 (x + 2) = 2 x \end{array}

\begin{array}{c} 12 \cdot \frac{2 x - 1}{3} - 12 \cdot \frac{x + 2}{4} \\ = 12 \cdot \frac{x}{6} \\ 4 (2 x - 1) - 3 (x + 2) = 2 x \end{array}

Expand the brackets

Distribute each multiplier over its whole bracket. The −3 multiplies both x and +2, giving −3x − 6; writing −3x + 6 is the most common slip at this point.

8 x - 4 - 3 x - 6 = 2 x

Collect like terms

Combine the x terms and the constants on the left: 8x − 3x = 5x and −4 − 6 = −10.

5 x - 10 = 2 x

Get the variable on one side

Subtract 2x from both sides, then add 10 to both sides. Each move applies the same operation to both sides, so the solutions do not change.

3 x = 10

Divide by the coefficient

Divide both sides by 3. The exact answer is a fraction; keep it as 10/3 rather than rounding it to 3.33.

x = \frac{10}{3}

Check in the original equation

Substitute 10/3 into the original equation. On the left, 2(10/3) − 1 = 17/3 and 10/3 + 2 = 16/3, so the left side is 17/9 − 4/3 = 5/9. On the right, (10/3)/6 = 5/9. The sides agree.

\begin{aligned} Left: & \frac{17}{9} - \frac{4}{3} = \frac{5}{9} \\ Right: & \frac{10}{3} \div 6 = \frac{5}{9} \end{aligned}

Result

x = 10/3. Substituting it gives 5/9 on both sides of the original equation.

x = \frac{10}{3}

Your turn

Solve 3(x − 2) + 4 = 5x − 8.

Show the answer and explanation

x = 3.

Expand and combine: 3x − 2 = 5x − 8. Subtracting 3x from both sides keeps the x coefficient positive: −2 = 2x − 8, so 6 = 2x and x = 3. Check: 3(1) + 4 = 7 and 5(3) − 8 = 7.

\begin{array}{l} 3 (x - 2) + 4 = 5 x - 8 \\ 3 x - 2 = 5 x - 8 \\ 6 = 2 x \\ x = 3 \end{array}

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Open the steps in Math, then change x/6 on the right to x/4 and solve again: the checker marks each new line you write.

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OpenStax College Algebra 2e — Linear equations in one variable

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