Operations that undo each other
Understand a variable, follow a calculation, and undo it in reverse order.
Interpret a variable as an unknown number.
Identify additive and multiplicative inverses.
Undo a sequence of operations in reverse order.
Before we begin
You’ll use: Add and subtract signed numbers. Multiply and divide whole numbers.
Try this quick warm-up. If it feels unfamiliar, follow the hint or revisit the previous lesson.
Calculate −3 + 8.
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Start with the idea
A variable such as x is a name for a number. In an equation we want to find a value that makes the statement true. In x + 4 = 11, x is the number that becomes 11 after adding 4.
Think of a calculation as a machine. It accepts an input, applies operations in a particular order, and gives an output. If we know the output, we can work backwards.
Putting on socks and then shoes is undone by removing shoes and then socks. Calculations follow the same reverse-order logic. A machine that multiplies by 3 and then adds 2 is undone by subtracting 2 first, then dividing by 3.
Inverse operations undo each other on their valid domains. Addition of is undone by subtraction of . Multiplication by a nonzero number is undone by division by . Division by zero is undefined. For , the expression is undone by subtracting and then dividing by .
Make the idea move
The final operation is the first one to undo. Try a negative input: does the same method still work?
Three examples, step by step
Undo addition
- The machine adds 6 and outputs 14: .
- Undo +6 with −6: .
- Check the input: .
Undo multiplication
- Four identical tickets cost £28: .
- Undo multiplication by 4 by dividing by 4: .
- Each ticket costs £7. Check: .
Reverse two operations
- The machine multiplies by 5, then subtracts 3, and outputs 17: .
- Undo −3 first: .
- Undo ×5 next: . Check: .
Your turn, with a little guidance
A machine multiplies an input by 3, then adds 4. Its output is 22. What was the input?
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Independent practice
Work on paper if it helps. Give a number, decimal, or fraction such as 3/2. Correct answers to every independent exercise complete this lesson; placement mastery requires a separate assessment.
Find x: x + 9 = 15.
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Find x: 6x = 42.
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Find x: 4x + 3 = 23.
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Find x: 2x − 7 = 11.
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Easy mistakes. Useful lessons.
The final operation was +4. Undo it first, or divide every term on both sides correctly. Dividing only the 22 changes the equation.
Multiplying by zero loses the input: every input becomes zero. There is no unique inverse.
The knowledge check
Close the examples and explain each point below in your own words. Then solve one practice question again without a hint.
- Variables name numbers. A solution makes an equation true.
- Inverse operations undo one another.
- Undo a calculation in reverse order, then check by substitution.
If you cannot explain a step yet, return to the visualization and change one value. What stays the same, and why?