Inverse operations

Inverse operations undo each other: + undoes −, and × undoes ÷. They are how you solve equations. A quiz for kids, parent help below.

🧒 Kids: undo it!

Adding 7 then taking away 7 gets you back where you started. That is an inverse! Use inverses to get x on its own. Then try the quiz.

▶ Quiz 🔢 More practice

22− 715÷ 35

💡 Trick: 3x + 7 = 22: take away the 7 first, then divide by 3. x = 5!

More equations: What is an equation · Variables · Algebraic expressions · Evaluating expressions · Writing expressions · Inverse operations

For parents and teachers ↓

This part is for grown-ups: the key idea, pictures to draw, what to say, common mistakes, questions with answers and a free worksheet.

The trick: undo the last step first

Every operation has an inverse that undoes it. To get x on its own, undo what was done to x, starting with the last operation. Do each inverse to both sides of the equation.

Patterns to spot

Key words

Why it matters: Inverse operations are the engine of solving equations. Every equation, from x + 4 = 9 to 3(x − 2) = 4x + 1, is solved by undoing operations in reverse order on both sides.

See it: pictures to draw with your child

Draw these on paper or a whiteboard. A quick sketch makes the idea concrete before any rules.

5× 315+ 722
Forwards: start with x = 5, multiply by 3, add 7. You get 22.
22− 715÷ 35
Backwards: undo + 7 with − 7, then undo × 3 with ÷ 3. Back to 5.
xx
2x + 3 = 11: take 3 blocks off both sides, then split into 2 groups. x = 4.

What your child already knows

Children already know fact families: 4 + 5 = 9 means 9 − 5 = 4, and 3 × 6 = 18 means 18 ÷ 6 = 3. Inverse operations are fact families used on purpose to find an unknown.

In real life: Working backwards is everyday thinking. If you spent $7 and have $15 left, you started with 15 + 7 = $22. If 3 friends split a bill and each paid $8, the bill was 3 × 8 = $24.

Undo in reverse order

If you put on socks and then shoes, you take off the shoes first. Equations work the same way: undo the last operation first.

In 3x + 7 = 22, x was multiplied by 3 and then 7 was added. So undo the + 7 first (subtract 7) and the × 3 last (divide by 3).

A function machine shows this clearly: run it forwards to build the equation, then backwards with the inverse operations to solve it.

Brackets change the order. In 3(x + 2) = 21, the + 2 happens first inside the brackets, so it is undone last: divide by 3 to get x + 2 = 7, then subtract 2.

Worked examples

Example 1: One step

  1. Solve x − 6 = 10.
  2. The 6 was subtracted, so add 6 to both sides: x = 16.

Example 2: Multiply

  1. Solve 5x = 35. x was multiplied by 5, so divide both sides by 5: x = 7.

Example 3: Two steps

  1. Solve 4x + 3 = 23.
  2. Undo + 3 first: 4x = 20.
  3. Then undo × 4: x = 5.

Example 4: Divide then subtract

  1. Solve x ÷ 2 − 4 = 6.
  2. Undo − 4 first: x ÷ 2 = 10.
  3. Then undo ÷ 2: x = 20.
  4. Check: 20 ÷ 2 − 4 = 6.

Common mistakes

Using the same operationSolves x + 5 = 12 by adding 5.Fix: the inverse of + 5 is − 5, so x = 7.
Undoing in the wrong orderDivides 3x + 7 = 22 by 3 first.Fix: undo the + 7 first, then divide.
One side onlySubtracts 5 on the left but not on the right.Fix: do every inverse to both sides.

What to say to your child

A short script for your first session. Pause after each line and let your child answer.

  1. “What undoes + 7?” (− 7.)
  2. “What undoes × 3?” (÷ 3.)
  3. “In 3x + 7 = 22, what was done to x first?” (× 3.)
  4. “So what do we undo first?” (The + 7.)
  5. “What is x?” (5.)

Questions to try (answers below)

Ask these out loud or write them down. The last one is a stretch.

  1. What undoes − 8?
  2. What undoes ÷ 5?
  3. x + 11 = 20
  4. x − 4 = 13
  5. 6x = 42
  6. x ÷ 3 = 9
  7. First step for 2x + 5 = 17?
  8. 2x + 5 = 17
  9. Stretch: (x − 3) ÷ 2 = 5
Show the answers
  1. + 8
  2. × 5
  3. 9
  4. 17
  5. 7
  6. 27
  7. Subtract 5
  8. 6
  9. 13

Games to play at home

Number machine

Your child secretly picks a number, does “× 2 + 3” and tells you the result. You undo it to find their number.

Rewind the day

Describe getting dressed, then list the undressing steps in reverse order.

Fact family triangles

Draw triangles with 3 numbers and say all four facts.

A 5-day plan for inverse operations

Ten minutes a day is enough. Short, regular practice works far better than one long session.

By grade

Grade 6 (Common Core 6.EE.B.7) solves one-step equations with inverse operations. Grade 7 (7.EE.B.4a) uses them in two steps, and grade 8 adds square roots. In India, this is the heart of the class 7 chapter “Finding the Unknown”.

More for each grade: Grade 6 math and the CBSE class 6 maths guide.

What comes next

Next, put inverses to work in one-step equations, then check steps with the equation solver.

Worksheet and more practice

Print a free worksheet with an answer key, or keep practicing online with equation practice.

Printable worksheets: evaluating expressions · combining like terms · one-step equations · make a fresh equations test · make an expressions test

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More: solving equations for kids · equation solver · balance scale · equation practice · order of operations · integers

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Questions parents ask

What are inverse operations?

Operations that undo each other. Addition and subtraction are inverses, and so are multiplication and division.

What is the inverse of multiplying by 4?

Dividing by 4.

How do inverse operations help solve equations?

They undo what was done to x, one step at a time, until x is on its own.

Which step do you undo first?

The last one that was done. In 3x + 7 = 22, undo the + 7 before the × 3.

What grade learns inverse operations?

Grade 6 under Common Core 6.EE.B.7, building on fact families from grades 1 to 3. In India, class 7.