Algebra Foundations

Two-Step Equations

Solving equations that need two inverse operations.

Foundational15 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

One-Step Equations

When One Step Isn't Enough

Not every equation isolates the variable after a single move. In an equation like 3x + 4 = 19, the variable x has two things attached to it: it's multiplied by 3, and then 4 is added to the result. Solving it takes two inverse operations, applied in a specific order.

The order matters, and it's the reverse of the order you'd use to build the expression in the first place. To evaluate 3x + 4 for a given x, you'd multiply first, then add. To undo it and solve for x, you work backward: undo the addition first, then undo the multiplication.

Worked Example — Solving a Two-Step Equation

Solve 3x + 4 = 19. Step 1: undo the addition by subtracting 4 from both sides: 3x + 4 − 4 = 19 − 4, which simplifies to 3x = 15. Step 2: undo the multiplication by dividing both sides by 3: 3x ÷ 3 = 15 ÷ 3, which simplifies to x = 5. Check: 3(5) + 4 = 15 + 4 = 19. True.

Worked Example — A Two-Step Equation with Subtraction

Solve 2x − 7 = 9. Step 1: undo the subtraction by adding 7 to both sides: 2x − 7 + 7 = 9 + 7, which simplifies to 2x = 16. Step 2: undo the multiplication by dividing both sides by 2: x = 8. Check: 2(8) − 7 = 16 − 7 = 9. True.

3x + 4 = 19 \;\Longrightarrow\; 3x = 15 \;\Longrightarrow\; x = 5

(1)

Equation Editor

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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

It can help to think of the variable as being wrapped in layers, like a present — the addition or subtraction term is the outer layer, and the multiplication or division is the inner layer. Unwrap the outside first.

Common Mistakes

Dividing by the coefficient before removing the added or subtracted number, such as solving 3x + 4 = 19 by immediately dividing everything by 3.
Undo addition or subtraction first, then undo multiplication or division last — the reverse of the order of operations used to build the expression.
Forgetting to apply the second step to both sides after already simplifying one side, such as stopping at 3x = 15 without dividing.
A two-step equation isn't solved until the variable is completely alone on one side. After the first inverse operation, check whether the variable still has a coefficient attached, and undo that too.

Key Takeaways

A two-step equation requires undoing addition or subtraction first, then multiplication or division.
Work backward from the order the expression was built in.
A solution isn't finished until the variable stands alone — always check by substituting back into the original equation.

Summary

Two-step equations build directly on one-step equations by chaining two inverse operations in the right order. This same balance-and-undo approach scales up to the multi-step and more complex equations you'll meet in later units.

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