Multi-Step Equations: Steps and Examples

Multi-step equations take more than two steps to solve. First simplify each side: use the distributive property to remove parentheses, then combine like terms. Then solve the two-step equation that is left. For example, $2(x + 3) + 4x = 30$ simplifies to $6x + 6 = 30$ , so $x = 4$ .

Before this lesson: You should know how to solve two-step equations, use the distributive property and combine like terms.

Extra practice: The Algebra 1 Unit 2 Practice Pack has printable problems, answer keys and a cheat sheet for this unit.

What Is a Multi-Step Equation?

A multi-step equation has extra parts that make it look harder, like parentheses or several terms with the same variable. For example, $2(x + 3) + 4x = 30$ has parentheses and two $x$ -terms.

The trick is to clean up first. Once each side is simplified, the equation turns into a two-step equation you already know how to solve.

Four boxes in a row joined by arrows: 1. Simplify each side (distribute, combine like terms). 2. Undo adding or subtracting. 3. Undo multiplying or dividing. 4. Check. — Simplify first. Then it’s a two-step equation.

How to Solve Multi-Step Equations

Use the distributive property to remove any parentheses.
Combine like terms on each side.
Undo adding or subtracting, on both sides.
Undo multiplying or dividing, on both sides.
Check your answer in the original equation.

\begin{aligned} 2(x + 3) + 4x &= 30 \\ 2x + 6 + 4x &= 30 & &\text{distribute} \\ 6x + 6 &= 30 & &\text{combine like terms} \\ 6x &= 24 & &\text{subtract 6} \\ x &= 4 & &\text{divide by 6} \end{aligned}

Check: $2(4 + 3) + 4(4) = 14 + 16 = 30$ . It works.

Worked Examples

Example 1: Combine like terms first

Solve $4x + 3x - 5 = 23$ .

7x - 5 = 23

Step 1. Combine $4x$ and $3x$ (see combining like terms).

7x - 5 + 5 = 23 + 5

Step 2. Add 5 to both sides.

7x = 28

Step 3. Simplify.

\frac{7x}{7} = \frac{28}{7}

Step 4. Divide both sides by 7.

x = 4

Step 5. Simplify.

Answer: $x = 4$

Check: $4(4) + 3(4) - 5 = 16 + 12 - 5 = 23$ . It works.

Example 2: Parentheses

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

3x - 6 = 21

Step 1. Distribute the 3.

3x - 6 + 6 = 21 + 6

Step 2. Add 6 to both sides.

3x = 27

Step 3. Simplify.

\frac{3x}{3} = \frac{27}{3}

Step 4. Divide both sides by 3.

x = 9

Step 5. Simplify.

Answer: $x = 9$ . Shortcut: 21 divides evenly by 3, so you could divide both sides by 3 first. That gives $x - 2 = 7$ , so $x = 9$ . Both ways work.

Check: $3(9 - 2) = 3(7) = 21$ . It works.

Example 3: Distribute a negative, then combine

Solve $-4(2y - 1) + 3y = 39$ .

-8y + 4 + 3y = 39

Step 1. Distribute $-4$ . A negative times a negative is positive, so $(-4)(-1) = 4$ . (See multiplying and dividing integers.)

-5y + 4 = 39

Step 2. Combine $-8y$ and $3y$ .

-5y + 4 - 4 = 39 - 4

Step 3. Subtract 4 from both sides.

-5y = 35

Step 4. Simplify.

\frac{-5y}{-5} = \frac{35}{-5}

Step 5. Divide both sides by $-5$ .

y = -7

Step 6. Simplify.

Answer: $y = -7$

Check: $-4(2(-7) - 1) + 3(-7) = -4(-15) - 21 = 60 - 21 = 39$ . It works.

Example 4: Clear the fractions

Solve $\frac{x}{2} + \frac{x}{3} = 10$ .

6 \cdot \frac{x}{2} + 6 \cdot \frac{x}{3} = 6 \cdot 10

Step 1. The denominators are 2 and 3. Their least common multiple is 6. Multiply every term on both sides by 6.

3x + 2x = 60

Step 2. Simplify each product. $6 \cdot \frac{x}{2} = 3x$ and $6 \cdot \frac{x}{3} = 2x$ .

5x = 60

Step 3. Combine like terms.

\frac{5x}{5} = \frac{60}{5}

Step 4. Divide both sides by 5.

x = 12

Step 5. Simplify.

Answer: $x = 12$

Check: $\frac{12}{2} + \frac{12}{3} = 6 + 4 = 10$ . It works.

Example 5: Word problem

The sum of three consecutive integers is 72. What are the integers?

n + (n + 1) + (n + 2) = 72

Step 1. Consecutive integers come one after another, like 5, 6, 7. Let $n$ = the first integer. The next two are $n + 1$ and $n + 2$ . Their sum is 72.

3n + 3 = 72

Step 2. Combine like terms.

3n + 3 - 3 = 72 - 3

Step 3. Subtract 3 from both sides.

3n = 69

Step 4. Simplify.

\frac{3n}{3} = \frac{69}{3}

Step 5. Divide both sides by 3.

n = 23

Step 6. Simplify.

Step 7. Answer the question that was asked. The integers are $n = 23$ , $n + 1 = 24$ and $n + 2 = 25$ .

Answer: 23, 24 and 25.

Check: $23 + 24 + 25 = 72$ .

Common Mistakes

Mistake	What it looks like	How to fix it
Distributing to only the first term	$3(x - 2) = 21$ becomes $3x - 2 = 21$	Multiply both terms: $3x - 6 = 21$ .
Dropping a sign when distributing a negative	$-4(2y - 1)$ becomes $-8y - 4$	$(-4)(-1) = +4$ , so it is $-8y + 4$ .
Combining unlike terms	$2x + 6$ becomes $8x$	$2x$ and 6 are not like terms. Leave them as $2x + 6$ .
Not multiplying every term	$\frac{x}{2} + \frac{x}{3} = 10$ becomes $3x + 2x = 10$	Multiply the 10 by 6 too: $3x + 2x = 60$ .
Stopping too early	Answering $n = 23$ when the question asks for three integers	Reread the question. Here the answer is 23, 24 and 25.

Quick check: Substitute your answer into the original equation, not a simplified one. That way you also catch mistakes made while simplifying.

Practice Problems

Solve each equation. Check your answer.

$5x - 2x + 4 = 19$
$4(a + 2) = 28$
$6 - 2(b - 1) = 14$
$3(2m - 5) + m = 20$
$-2(3 - k) + 4k = 18$
$0.4(x + 10) = 6$
$\frac{x}{4} + \frac{x}{2} = 9$
$\frac{2(y + 1)}{3} = 6$
$5(p - 1) - 3(p + 2) = 7$
A rectangle’s length is 4 meters more than twice its width. Its perimeter is 50 meters. Find the width and the length.

Answers

Show answers

3x + 4 = 19

Combine like terms.

3x = 15

Subtract 4 from both sides.

x = 5

Divide both sides by 3.

4a + 8 = 28

Distribute the 4.

4a = 20

Subtract 8 from both sides.

a = 5

Divide both sides by 4.

6 - 2b + 2 = 14

Distribute the −2.

8 - 2b = 14

Combine the constants.

-2b = 6

Subtract 8 from both sides.

b = -3

Divide both sides by −2.

6m - 15 + m = 20

Distribute the 3.

7m - 15 = 20

Combine like terms.

7m = 35

Add 15 to both sides.

m = 5

Divide both sides by 7.

-6 + 2k + 4k = 18

Distribute the −2.

-6 + 6k = 18

Combine like terms.

6k = 24

Add 6 to both sides.

k = 4

Divide both sides by 6.

x + 10 = 15

Divide both sides by 0.4 (see operations with decimals)

x = 5

Subtract 10 from both sides.

4 \cdot \frac{x}{4} + 4 \cdot \frac{x}{2} = 4 \cdot 9

The least common multiple of 4 and 2 is 4. Multiply every term by 4.

x + 2x = 36

Simplify each product.

3x = 36

Combine like terms.

x = 12

Divide both sides by 3.

2(y + 1) = 18

Multiply both sides by 3.

y + 1 = 9

Divide both sides by 2.

y = 8

Subtract 1 from both sides.

5p - 5 - 3(p + 2) = 7

Distribute the 5.

5p - 5 - 3p - 6 = 7

Distribute the −3.

2p - 11 = 7

Combine like terms.

2p = 18

Add 11 to both sides.

p = 9

Divide both sides by 2.

10.

2w + 2(2w + 4) = 50

Let $w$ = the width, so the length is $2w + 4$ . Perimeter is 2 widths plus 2 lengths.

2w + 4w + 8 = 50

Distribute the 2.

6w + 8 = 50

Combine like terms.

6w = 42

Subtract 8 from both sides.

w = 7

Divide both sides by 6.

2(7) + 4 = 14 + 4 = 18

Find the length.

The width is 7 meters and the length is 18 meters.

Want more practice? The Algebra 1 Unit 2 Practice Pack has 51 pages of problems on this and the other lessons in the unit, with step-by-step answers and a cheat sheet.

Multi-Step Equations FAQ

What is a multi-step equation?

It is an equation that takes more than two steps to solve, usually because it has parentheses or like terms to combine first.

How do you solve multi-step equations?

Distribute to remove parentheses, combine like terms on each side, then undo adding or subtracting and multiplying or dividing. Check your answer at the end.

Do you always have to distribute first?

No. If the number outside the parentheses divides evenly into the other side, you can divide first. In $3(x - 2) = 21$ , dividing by 3 gives $x - 2 = 7$ right away.

How do you solve multi-step equations with fractions?

Multiply every term on both sides by the least common multiple of the denominators. This clears the fractions, so you can solve as usual.

Summary

Simplify each side first: distribute, then combine like terms.
After simplifying, solve the two-step equation that is left.
Distribute negative numbers carefully and watch the signs.
To clear fractions, multiply every term by the least common multiple of the denominators.
Check your answer in the original equation.

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Next lesson: Equations with Variables on Both Sides

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