Inverse Operations Formula

The Formula

a + b - b = a, \quad a \times b \div b = a \;(b \neq 0)

When to use: Adding 5 then subtracting 5 brings you back to where you started.

Quick Example

(x + 7) - 7 = x Start at x, add 7, subtract 7, back to x.

Notation

+ and - are inverse pairs; \times and \div are inverse pairs

What This Formula Means

Operations that undo each other: addition undoes subtraction, multiplication undoes division, and vice versa. Applying an operation followed by its inverse returns you to the starting value.

Adding 5 then subtracting 5 brings you back to where you started.

Formal View

\forall a, b \in \mathbb{R}: (a + b) - b = a; \; \forall b \neq 0: (a \cdot b) \div b = a

Worked Examples

Example 1

easy
Show that adding and subtracting the same number leaves you where you started: start with 15, add 8, then subtract 8. What do you get?

Solution

  1. 1
    Start: 15.
  2. 2
    Add 8: \(15 + 8 = 23\).
  3. 3
    Subtract 8: \(23 - 8 = 15\).
  4. 4
    You are back to 15. This shows \(a + b - b = a\).

Answer

15 (back to the start)
Addition and subtraction are inverse operations. Adding then subtracting the same number always returns you to the original value.

Example 2

medium
You start with a mystery number. You multiply it by 6 and get 54. Use the inverse operation to find the mystery number.

Common Mistakes

  • Applying the wrong inverse — using subtraction to undo multiplication instead of division
  • Thinking that the inverse of squaring is dividing by 2 instead of taking the square root
  • Forgetting that inverse operations must be applied to both sides of an equation

Why This Formula Matters

Inverse operations are the foundation of equation-solving — to isolate a variable, you apply the inverse of whatever operation acts on it.

Frequently Asked Questions

What is the Inverse Operations formula?

Operations that undo each other: addition undoes subtraction, multiplication undoes division, and vice versa. Applying an operation followed by its inverse returns you to the starting value.

How do you use the Inverse Operations formula?

Adding 5 then subtracting 5 brings you back to where you started.

What do the symbols mean in the Inverse Operations formula?

+ and - are inverse pairs; \times and \div are inverse pairs

Why is the Inverse Operations formula important in Math?

Inverse operations are the foundation of equation-solving — to isolate a variable, you apply the inverse of whatever operation acts on it.

What do students get wrong about Inverse Operations?

Squaring and square root are inverses (mostly—watch for \pm).

What should I learn before the Inverse Operations formula?

Before studying the Inverse Operations formula, you should understand: addition, subtraction, multiplication, division.

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