1Cademy - Solving $$8n - 4 = -2n + 6$$ by Collecting Variables and Constants

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Solving $8n - 4 = -2n + 6$ by Collecting Variables and Constants

To solve $8n - 4 = -2n + 6$ , notice that variable terms ( $8n$ and $-2n$ ) and constant terms ( $-4$ and $6$ ) appear on both sides of the equation. Because $8 > -2$ , 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: The term $-2n$ is on the constant side, so add $2n$ to both sides using the Addition Property of Equality:

$8n + 2n - 4 = -2n + 2n + 6$

Combine like terms: $10n - 4 = 6$ . Now the variable appears only on the left.

Step 2 — Remove the constant from the variable side: The constant $-4$ is on the variable side, so add $4$ to both sides:

$10n - 4 + 4 = 6 + 4$

$10n = 10$

Step 3 — Isolate the variable using the Division Property of Equality: Because $n$ is multiplied by the coefficient $10$ , divide both sides by $10$ :

$\frac{10n}{10} = \frac{10}{10}$

$n = 1$

Step 4 — Check by substitution: Replace $n$ with $1$ in the original equation:

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

$8 - 4 \stackrel{?}{=} -2 + 6$

$4 = 4 \checkmark$

Because both sides are equal, $n = 1$ is confirmed as the correct solution. Unlike the earlier examples that used the Subtraction Property to collect terms, this equation requires the Addition Property in both collecting steps — adding $2n$ to eliminate the negative variable term from the right side, and adding $4$ to eliminate the negative constant from the left side. Additionally, a final division step is needed because the combined coefficient ( $10$ ) is not equal to $1$ .

Updated 2026-04-21

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OpenStax Elementary Algebra

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