1Cademy - Solving $$7a - 3 = 13a + 7$$ by Collecting Variables and Constants

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Solving $7a - 3 = 13a + 7$ by Collecting Variables and Constants

To solve $7a - 3 = 13a + 7$ , observe that variable terms ( $7a$ and $13a$ ) and constant terms ( $-3$ and $7$ ) appear on both sides of the equation. Because $13 > 7$ , designate the right side as the variable side and the left side as the constant side.

Step 1 — Remove the variable term from the constant side: Since $7a$ is on the constant side, subtract $7a$ from both sides using the Subtraction Property of Equality:

$7a - 7a - 3 = 13a - 7a + 7$

Combine like terms: $-3 = 6a + 7$ . Now the variable appears only on the right.

Step 2 — Remove the constant from the variable side: The constant $7$ is on the variable side, so subtract $7$ from both sides:

$-3 - 7 = 6a + 7 - 7$

$-10 = 6a$

Step 3 — Isolate the variable using the Division Property of Equality: Because $a$ is multiplied by the coefficient $6$ , divide both sides by $6$ :

$\frac{-10}{6} = \frac{6a}{6}$

Simplify the fraction by dividing numerator and denominator by their greatest common factor $2$ :

$-\frac{5}{3} = a$

Step 4 — Check by substitution: Replace $a$ with $-\frac{5}{3}$ in the original equation:

$7\left(-\frac{5}{3}\right) - 3 \stackrel{?}{=} 13\left(-\frac{5}{3}\right) + 7$

$-\frac{35}{3} - \frac{9}{3} \stackrel{?}{=} -\frac{65}{3} + \frac{21}{3}$

$-\frac{44}{3} = -\frac{44}{3} \checkmark$

Because both sides are equal, $a = -\frac{5}{3}$ is confirmed as the correct solution. This example differs from earlier ones in two important ways: (1) the right side is chosen as the variable side because its coefficient ( $13$ ) is larger, demonstrating that either side can serve as the variable side; and (2) the final division does not produce a whole number, so the answer must be expressed as a simplified fraction.

Updated 2026-04-21

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OpenStax Elementary Algebra

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