How to Solve Multi-Step Equations

Quick answer

To solve a multi-step equation, first simplify each side on its own: distribute to clear parentheses, then combine like terms. Once each side is a single simplified expression, undo the operations in reverse order exactly as you would for a two-step equation. Always check by substituting your answer back in.

What you'll learn

Simplify each side of an equation before solving
Apply the distributive property to clear parentheses
Combine like terms correctly, keeping signs attached

What a multi-step equation is

A multi-step equation needs more than two operations because one or both sides must be simplified before you can start undoing anything:

4(x + 3) - 2x = 18

You cannot subtract or divide yet — the left side is not in the form “something times $x$ , plus a number”. Getting it into that form is the new work.

Why you simplify before you solve

A two-step equation like $3x + 7 = 22$ works because the left side is already a clean chain of operations on $x$ : multiply, then add. You can walk that chain backwards.

$4(x + 3) - 2x$ is not a chain — it is a tangle. There are two separate terms containing $x$ , and one of them is trapped inside parentheses. Before you can reverse anything, you need a single $x$ -term and a single constant.

So the job splits into two phases:

Simplify each side until it looks like $ax + b$ .
Solve it exactly as you already know how.

Phase 2 is nothing new. Everything hard about multi-step equations lives in phase 1.

Watching one equation move through both phases shows where the split falls:

Line	The move made	Which phase
$4(x + 3) - 2x = 18$	the starting tangle	—
$4x + 12 - 2x = 18$	distributed	simplify
$2x + 12 = 18$	combined like terms	simplify
$2x = 6$	subtracted $12$	solve
$x = 3$	divided by $2$	solve

By line three the problem has become an ordinary two-step equation. The last two lines are a lesson you already finished.

Fractions, and how to make them leave

An equation full of fractions looks worse than it is, because there is a single move that removes all of them at once: multiply every term on both sides by the least common denominator.

\frac{x}{3} + \frac{1}{2} = \frac{5}{6}

The denominators are $3$ , $2$ and $6$ , so the least common denominator is $6$ . Multiplying every term by $6$ :

6 \cdot \frac{x}{3} + 6 \cdot \frac{1}{2} = 6 \cdot \frac{5}{6}

2x + 3 = 5

The fractions are gone in one step, and what remains is a two-step equation giving $x = 1$ .

How to solve a multi-step equation

Distribute to remove parentheses.
Combine like terms on each side.
Undo addition or subtraction on both sides.
Undo multiplication or division on both sides.
Check by substituting into the original equation.

The distributive property

a(b + c) = ab + ac

The multiplier outside touches every term inside. Watch the signs — a negative outside flips the sign of everything it reaches:

-2(x - 5) = -2x + 10

Worked examples

Example 3 — Two sets of parentheses

Solve $2(x + 1) + 3(x - 2) = 14$ .

Step 1 — Distribute both.

2x + 2 + 3x - 6 = 14

Step 2 — Combine like terms. $2x + 3x = 5x$ and $2 - 6 = -4$ .

5x - 4 = 14

Step 3 and 4 — Add $4$ , then divide by $5$ .

5x = 18 \quad\text{so}\quad x = \frac{18}{5}

Check.

2\left(\frac{18}{5} + 1\right) + 3\left(\frac{18}{5} - 2\right) = 2\left(\frac{23}{5}\right) + 3\left(\frac{8}{5}\right) = \frac{46}{5} + \frac{24}{5} = \frac{70}{5} = 14 \;\checkmark

A fraction is a perfectly good answer. Do not assume you made a mistake.

Common mistakes

Practice problems

Solve $3(x + 2) = 21$ .

Hint

Both sides divide evenly by $3$ , so you can skip distributing.

Answer

$x = 5$

Full solution

Divide both sides by $3$ : $x + 2 = 7$ . Subtract $2$ : $x = 5$ .

Check: $3(5 + 2) = 21$ ✓
Solve $5(x - 1) + 2 = 17$ .

Answer

$x = 4$

Full solution

Distribute: $5x - 5 + 2 = 17$ . Combine: $5x - 3 = 17$ .

Add $3$ : $5x = 20$ . Divide by $5$ : $x = 4$ .

Check: $5(4 - 1) + 2 = 15 + 2 = 17$ ✓
Solve $2x + 3x - 7 = 23$ .

Answer

$x = 6$

Full solution

Combine: $5x - 7 = 23$ . Add $7$ : $5x = 30$ . Divide by $5$ : $x = 6$ .

Check: $2(6) + 3(6) - 7 = 12 + 18 - 7 = 23$ ✓
Solve $-2(k + 3) = 10$ .

Answer

$k = -8$

Full solution

Distribute: $-2k - 6 = 10$ . Add $6$ : $-2k = 16$ . Divide by $-2$ : $k = -8$ .

Check: $-2(-8 + 3) = -2(-5) = 10$ ✓
Solve $4(n - 1) + 2(n + 3) = 26$ .

Hint

Distribute both parentheses before combining anything.

Answer

$n = 4$

Full solution

Distribute: $4n - 4 + 2n + 6 = 26$ . Combine: $6n + 2 = 26$ .

Subtract $2$ : $6n = 24$ . Divide by $6$ : $n = 4$ .

Check: $4(3) + 2(7) = 12 + 14 = 26$ ✓
Solve $8 - 3(x - 2) = 5$ .

Hint

The $-$ in front belongs to the $3$ . Distribute $-3$ .

Answer

$x = 3$

Full solution

Distribute: $8 - 3x + 6 = 5$ . Combine: $-3x + 14 = 5$ .

Subtract $14$ : $-3x = -9$ . Divide by $-3$ : $x = 3$ .

Check: $8 - 3(3 - 2) = 8 - 3 = 5$ ✓
Solve $3(2x + 1) - 4x = 11$ .

Answer

$x = 4$

Full solution

Distribute: $6x + 3 - 4x = 11$ . Combine: $2x + 3 = 11$ .

Subtract $3$ : $2x = 8$ . Divide by $2$ : $x = 4$ .

Check: $3(9) - 16 = 27 - 16 = 11$ ✓
A rectangle is $3$ cm longer than it is wide. Its perimeter is $26$ cm. Find the width.

Hint

Perimeter is $2(\text{width}) + 2(\text{length})$ , and length $=$ width $+ 3$ .

Answer

Width $= 5$ cm

Full solution

Let $w$ be the width, so the length is $w + 3$ .

$2w + 2(w + 3) = 26$

Distribute: $2w + 2w + 6 = 26$ . Combine: $4w + 6 = 26$ .

Subtract $6$ : $4w = 20$ . Divide by $4$ : $w = 5$ .

Check: width $5$ , length $8$ , perimeter $2(5) + 2(8) = 26$ ✓

Frequently asked questions

Do I always have to distribute first?

No. If the whole parenthesis is multiplied by a number that divides the other side evenly, dividing first is faster. For 4(x + 3) = 20, dividing both sides by 4 gives x + 3 = 5 immediately. Distributing works every time, though, so it is the safer default.

What counts as a like term?

Terms with exactly the same variable raised to exactly the same power. 3x and 5x are like terms; 3x and 3x squared are not; 3x and 7 are not. Only like terms can be added or subtracted.

What if there are fractions in the equation?

Multiply every term on both sides by the least common denominator first. That clears the fractions in one move and leaves an ordinary equation. Multiplying only some terms changes the equation, so nothing may be skipped.

Does the order of the steps matter?

The order given is the reliable one: distribute, combine like terms, then undo the operations. Other orders can work, but they usually create more chances to slip, and there is no reward for finding a shorter route to the same answer.

How do I know when I am finished?

When the variable stands alone on one side with a coefficient of one, and a number stands alone on the other. Anything else, such as 2x = 8, still has one step to go.

What to learn next

Key terms in this lesson

Distributive property: The distributive property says a(b + c) = ab + ac. The multiplier outside the parentheses multiplies every term inside, not only the first one.
Like terms: Like terms have exactly the same variables raised to exactly the same powers. 3x and 5x are like terms, but 3x and 3x squared are not, and neither are 3x and 7.
Term: A term is one piece of an expression, separated from the others by plus or minus signs. The expression 4x + 12 - 2x has three terms: 4x, 12 and -2x.

Standards alignment

This lesson covers the following Common Core State Standards for Mathematics.

CCSS.MATH.CONTENT.8.EE.C.7bExpressions and EquationsSolve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.

For teachersPrint problems onlyTick to print a blank worksheet; leave it clear to print the lesson with the full answer key.

Quick answer

What you'll learn

Frequently asked questions

Do I always have to distribute first?

What counts as a like term?

What if there are fractions in the equation?

Does the order of the steps matter?

How do I know when I am finished?

What to learn next

Key terms in this lesson

Standards alignment

More Algebra 1 lessons