How to Solve Two-Step Equations

Quick answer

To solve a two-step equation, undo the operations in reverse order: first undo any addition or subtraction, then undo the multiplication or division. For 3x + 7 = 22, subtract 7 from both sides to get 3x = 15, then divide both sides by 3 to get x = 5.

What you'll learn

Solve a two-step equation for an unknown variable
Explain why operations are undone in reverse order
Check a solution by substituting it back into the original equation

What a two-step equation is

A two-step equation takes two inverse operations to solve. The variable has been multiplied by something and had something added to it:

3x + 7 = 22

Read it as a set of instructions performed on $x$ : take $x$ , multiply by $3$ , then add $7$ , and the result is $22$ . Solving means running those instructions backwards.

Why you undo operations in reverse order

This is the part most courses skip, and it is the whole idea.

To build the left side you follow the order of operations: multiply first, then add.

x \;\xrightarrow{\;\times 3\;}\; 3x \;\xrightarrow{\;+7\;}\; 3x + 7

To undo it you reverse the arrows, and reversing a sequence means starting from the end:

3x + 7 \;\xrightarrow{\;-7\;}\; 3x \;\xrightarrow{\;\div 3\;}\; x

It is exactly like getting dressed. Socks go on before shoes, so shoes come off before socks. You cannot take your socks off first.

That is why you undo addition and subtraction before multiplication and division — the reverse of the order you would use to evaluate the expression.

What each step buys you

It helps to see that the two steps are not interchangeable chores. Each one has a job, and the order is what leaves the second job with nothing to fight.

After this step	The equation reads	What is left to do
start	$3x + 7 = 22$	two operations wrapped around $x$
subtract $7$	$3x = 15$	one operation — this is now a one-step equation
divide by $3$	$x = 5$	nothing

The first step does not solve anything. It converts the problem into one you already know how to finish, and that is the pattern every later lesson repeats: strip the outer layer, and what remains is a problem you have already met. See one-step equations for the finish, and multi-step equations for what happens when there are more layers.

When the numbers are not whole

Nothing in the method changes when fractions or negatives appear — only the arithmetic gets fussier.

-4x + 9 = 25 \quad\Rightarrow\quad -4x = 16 \quad\Rightarrow\quad x = -4

Subtracting $9$ from both sides is the first step whether the coefficient is $3$ or $-4$ . The negative only matters at the division, and it matters there because $\tfrac{16}{-4}$ is $-4$ , not $4$ . Carrying the sign is the entire difficulty, and checking catches it: $-4(-4) + 9 = 16 + 9 = 25$ ✓

How to solve a two-step equation

Undo the addition or subtraction. Do it to both sides.
Undo the multiplication or division. Do it to both sides.
Simplify.
Check by substituting your answer into the original equation.

Worked examples

Common mistakes

Practice problems

Solve $2x + 5 = 13$ .

Hint

Undo the $+5$ first, then the $\times 2$ .

Answer

$x = 4$

Full solution

Subtract $5$ : $2x = 8$ . Divide by $2$ : $x = 4$ .

Check: $2(4) + 5 = 13$ ✓
Solve $7y - 3 = 25$ .

Answer

$y = 4$

Full solution

Add $3$ : $7y = 28$ . Divide by $7$ : $y = 4$ .

Check: $7(4) - 3 = 25$ ✓
Solve $\tfrac{a}{2} + 6 = 10$ .

Answer

$a = 8$

Full solution

Subtract $6$ : $\tfrac{a}{2} = 4$ . Multiply by $2$ : $a = 8$ .

Check: $8 \div 2 + 6 = 10$ ✓
Solve $-3m + 11 = 2$ .

Answer

$m = 3$

Full solution

Subtract $11$ : $-3m = -9$ . Divide by $-3$ : $m = 3$ .

Check: $-3(3) + 11 = -9 + 11 = 2$ ✓
Solve $4x + 9 = 9$ .

Hint

Do not be thrown by the matching numbers. Follow the same two steps.

Answer

$x = 0$

Full solution

Subtract $9$ : $4x = 0$ . Divide by $4$ : $x = 0$ .

Check: $4(0) + 9 = 9$ ✓ Zero is a perfectly good solution.
Solve $6 - 2x = 14$ .

Hint

Rewrite as $-2x + 6 = 14$ so the terms are in the usual order.

Answer

$x = -4$

Full solution

Subtract $6$ : $-2x = 8$ . Divide by $-2$ : $x = -4$ .

Check: $6 - 2(-4) = 6 + 8 = 14$ ✓
Solve $\tfrac{k}{5} - 2 = -1$ .

Answer

$k = 5$

Full solution

Add $2$ : $\tfrac{k}{5} = 1$ . Multiply by $5$ : $k = 5$ .

Check: $5 \div 5 - 2 = 1 - 2 = -1$ ✓
A gym charges a $25 joining fee plus $15 per month. Ana paid $115 in total. For how many months did she pay?

Hint

Total $=$ monthly rate $\times$ months $+$ joining fee.

Answer

$6$ months

Full solution

Let $m$ be the number of months: $15m + 25 = 115$ .

Subtract $25$ : $15m = 90$ . Divide by $15$ : $m = 6$ .

Check: $15(6) + 25 = 90 + 25 = 115$ ✓

Frequently asked questions

Why do you subtract before you divide?

You are undoing the order of operations, so you work backwards. Order of operations does multiplication before addition, so undoing it does subtraction before division. Dividing first also works but forces you to divide every term, which usually creates fractions.

Can I divide first instead?

Yes, as long as you divide every term on both sides. For 3x + 7 = 22 you would get x + 7/3 = 22/3, which gives the same answer but with harder arithmetic. Subtracting first keeps the numbers whole.

What if the variable is on the right instead of the left?

Nothing changes. Solve 11 = 2x + 3 exactly as you would 2x + 3 = 11: subtract 3 from both sides, then divide by 2. You can also swap the sides first if it feels easier to read.

What do I do with a fraction in front of the variable?

Multiply both sides by its reciprocal, or by the bottom number. To undo two-thirds of x, multiply both sides by three halves. Multiplying is the inverse of dividing, so the fraction cancels.

What to learn next

How to Solve Multi-Step EquationsSolve multi-step equations by distributing, combining like terms, then undoing operations in reverse order, with worked examples and practice problems.

Key terms in this lesson

Coefficient: A coefficient is the number multiplying a variable in a term. In 3x the coefficient is 3, and in -5y it is -5, because the sign belongs to the number.
Constant: A constant is a term with a fixed value and no variable attached. In 3x + 7, the constant is 7, because it stays the same whatever x turns out to be.
Inverse operations: Inverse operations undo each other. Addition and subtraction are inverses, and so are multiplication and division. Solving an equation means applying the inverse of whatever was done to the variable.

Standards alignment

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

CCSS.MATH.CONTENT.7.EE.B.4aExpressions and EquationsSolve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach.

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