Linear Equations & Inequalities

Multi-Step Equations

Solving linear equations that require multiple steps, including the distributive property.

Foundational20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Combining Every Tool So Far

A multi-step equation is any equation that needs more than two moves to solve — often because it has parentheses to distribute, like terms to combine, or the variable appearing on both sides. The good news is that no new technique is required: multi-step equations just chain together the distributing, combining, and undoing you've already practiced, in a consistent order.

A reliable order to follow: first distribute to clear any parentheses, then combine like terms on each side separately, then move all variable terms to one side and all constant terms to the other, and finally divide to isolate the variable.

Worked Example — Distribute, Combine, Then Isolate

Solve 3(x + 2) − 4 = 11. Distribute first: 3x + 6 − 4 = 11. Combine the constants on the left: 3x + 2 = 11. Subtract 2 from both sides: 3x = 9. Divide by 3: x = 3. Check: 3(3 + 2) − 4 = 3(5) − 4 = 15 − 4 = 11.

Worked Example — A Variable on Both Sides

Solve 5x − 3 = 2x + 9. There's no parentheses to distribute, but the variable appears on both sides, so gather the x-terms on one side first: subtract 2x from both sides to get 3x − 3 = 9. Add 3 to both sides: 3x = 12. Divide by 3: x = 4. Check: 5(4) − 3 = 17, and 2(4) + 9 = 17. Both sides match.

Equation Editor

Edit as raw LaTeX

sum(i, 1, n, i^2) = n(n+1)(2n+1)/6

Constants

Structures

Calculus & discrete math

Greek

\sum_{i=1}^{n} i^{2} = \frac{\operatorname{n}\left(n + 1\right) \cdot \left(2 \cdot n + 1\right)}{6}

Evaluate

Unknown function "n"

Tip

When the variable shows up on both sides, subtracting the smaller variable term from both sides (rather than the larger one) keeps the remaining coefficient positive, which is usually easier to work with.

Common Mistakes

Combining like terms across the equal sign, such as treating 3x on the left and 2x on the right as if they were already combined.
Like terms can only be combined when they're on the same side of the equation. Terms on opposite sides need to be moved to the same side first, by adding or subtracting, before they can combine.
Distributing correctly but then forgetting to also combine the resulting like terms before isolating the variable.
After distributing, scan each side for terms that can still be combined. An equation is ready for the isolating steps only once each side is fully simplified.

Key Takeaways

Solve multi-step equations in a consistent order: distribute, combine like terms on each side, gather variable terms to one side, then isolate.
Variables on both sides can be combined once they're moved to the same side of the equation.
The same balance-and-undo logic from one- and two-step equations still applies — multi-step equations just chain more of it together.

Summary

Multi-step equations combine distributing, combining like terms, and isolating a variable — the complete toolkit for solving linear equations. Next, you'll apply this same process to equations that involve more than one variable.

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