1Cademy - Solving $$\frac{5}{4}x + 6 = \frac{1}{4}x - 2$$ by Collecting Variables and Constants

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Solving $\frac{5}{4}x + 6 = \frac{1}{4}x - 2$ by Collecting Variables and Constants

To solve $\frac{5}{4}x + 6 = \frac{1}{4}x - 2$ , observe that both sides contain a variable term with a fractional coefficient as well as a constant term. Because $\frac{5}{4} > \frac{1}{4}$ , designate the left side as the variable side and the right side as the constant side.

Step 1 — Remove the variable term from the constant side: Since $\frac{1}{4}x$ is on the constant side, subtract $\frac{1}{4}x$ from both sides using the Subtraction Property of Equality:

$\frac{5}{4}x - \frac{1}{4}x + 6 = \frac{1}{4}x - \frac{1}{4}x - 2$

Combine the like terms on the left: the fractional coefficients share the denominator $4$ , so $\frac{5}{4} - \frac{1}{4} = \frac{4}{4} = 1$ . The equation simplifies to $x + 6 = -2$ . Now the variable appears only on the left.

Step 2 — Remove the constant from the variable side: The constant $6$ is on the variable side, so subtract $6$ from both sides:

$x + 6 - 6 = -2 - 6$

$x = -8$

Because the coefficient of $x$ is already $1$ , no further division is needed.

Step 3 — Check by substitution: Replace $x$ with $-8$ in the original equation:

$\frac{5}{4}(-8) + 6 \stackrel{?}{=} \frac{1}{4}(-8) - 2$

-10 + 6 stackrel{?}{=} -2 - 2

-4 = -4 checkmark

Because both sides are equal, $x = -8$ is confirmed as the correct solution. This example demonstrates that the collecting technique works identically when the variable terms have fractional coefficients. When two fractions with the same denominator are subtracted — here $\frac{5}{4}x - \frac{1}{4}x$ — the numerators are combined over the common denominator, and the result may simplify to a whole number, making the remaining steps straightforward.

Updated 2026-04-21

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OpenStax Elementary Algebra

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