Solving Two-Step Equations

From one step to two

A one-step equation like x + 5 = 12 takes a single move to solve: subtract 5 and you get x = 7. A two-step equation needs two moves because two things have been done to the variable.

Look at 2x + 3 = 11. Here x has been multiplied by 2 and had 3 added. To find x we have to undo both operations, one at a time, in the right order. This lesson gives you a reliable routine that works every time. If you have not solved one-step equations yet, start with solving linear equations, then return here.

The balance rule (the golden rule)

Think of the = sign as the middle of a balance scale. Both sides weigh exactly the same. The rule never changes:

Whatever you do to one side, you must do to the other side.

If you only change one side, the scale tips and the equation stops being true. Every step below obeys this rule.

Inverse operations

To remove an operation we use its inverse — the operation that undoes it:

Operation	Its inverse
Add (+)	Subtract (−)
Subtract (−)	Add (+)
Multiply (×)	Divide (÷)
Divide (÷)	Multiply (×)

Solving an equation is just peeling these operations off the variable until it stands alone.

The order: reverse the order of operations

Here is the key idea that makes two-step equations easy. When you build an expression you follow the order of operations (multiply before you add). When you solve, you go in reverse:

Undo addition and subtraction first, then undo multiplication and division.

In 2x + 3, the last thing done to x was "+3", so that is the first thing we undo. Then we undo the "×2".

Worked example 1 — the classic

Solve 2x + 3 = 11.

Step 1 — Undo the +3. Subtract 3 from both sides. 2x + 3 − 3 = 11 − 3 2x = 8

Step 2 — Undo the ×2. Divide both sides by 2. 2x ÷ 2 = 8 ÷ 2 x = 4

Step 3 — Check. Put x = 4 back into the original: 2(4) + 3 = 8 + 3 = 11. ✓ Both sides match, so x = 4 is correct.

Worked example 2 — with subtraction

Solve 5x − 4 = 16.

Step 1 — Undo the −4. Add 4 to both sides. 5x − 4 + 4 = 16 + 4 5x = 20

Step 2 — Undo the ×5. Divide both sides by 5. x = 20 ÷ 5 = 4

Check: 5(4) − 4 = 20 − 4 = 16. ✓

Worked example 3 — a negative answer

Solve 3x + 7 = 1.

Step 1 — Subtract 7 from both sides. 3x = 1 − 7 = −6

Step 2 — Divide both sides by 3. x = −6 ÷ 3 = −2

Check: 3(−2) + 7 = −6 + 7 = 1. ✓ Do not be alarmed by negative answers — they are completely valid. If signs trip you up, revisit introduction to integers.

Worked example 4 — the variable is divided

Solve x/4 − 5 = 2.

Here x has been divided by 4 and then 5 was subtracted. Undo the subtraction first, then the division.

Step 1 — Add 5 to both sides. x/4 = 2 + 5 = 7

Step 2 — Undo the ÷4 by multiplying both sides by 4. x = 7 × 4 = 28

Check: 28/4 − 5 = 7 − 5 = 2. ✓

Worked example 5 — variable not in front

Solve 18 = 4x + 2. Having the number on the left is fine — the rules are identical.

Step 1 — Subtract 2 from both sides. 16 = 4x

Step 2 — Divide both sides by 4. 4 = x, so x = 4.

Check: 4(4) + 2 = 16 + 2 = 18. ✓

A summary of the routine

For any two-step equation, follow these four steps:

1. Identify the two operations attached to the variable.
2. Undo the +/− by doing the opposite to both sides.
3. Undo the ×/÷ by doing the opposite to both sides.
4. Check by substituting back into the original equation.

Equation	Step 1 (+/−)	Step 2 (×/÷)	Answer
2x + 3 = 11	2x = 8	x = 4	4
5x − 4 = 16	5x = 20	x = 4	4
x/4 − 5 = 2	x/4 = 7	x = 28	28
3x + 7 = 1	3x = −6	x = −2	−2

Why the order matters

Could you undo the multiplication first? Sometimes, but it gets messy. In 2x + 3 = 11, dividing everything by 2 first gives x + 1.5 = 5.5, which works but introduces awkward decimals. Removing the "+3" first keeps the numbers whole and clean. Undoing the addition or subtraction before the multiplication or division is the safe, reliable order — it mirrors unwrapping a parcel by taking off the outer layer first.

Practice activity

Make your own two-step equation puzzle for a partner.

1. Choose a secret value for x (say x = 5).
2. Multiply it by a number, then add or subtract another — e.g. 3 × 5 = 15, then + 4 = 19, giving the equation 3x + 4 = 19.
3. Hand the equation to a partner and challenge them to find your secret x by undoing the steps.
4. Both of you check the answer by substituting it back. Swap roles and repeat with negatives and fractions to build confidence.

Quick recap

A two-step equation needs two inverse operations. Undo the addition or subtraction first, then the multiplication or division, doing the same thing to both sides each time. Finish by substituting your answer back into the original equation to be sure it balances.