Key takeaway: Inverse operations are pairs of operations that undo each other, such as addition with subtraction and multiplication with division, and they are the main tool for solving equations.
Grade 66.EE.B.7Expressions and Equations
Definition of Inverse Operations
Inverse operations reverse each other. If you add 36 to a number and then subtract 36, you are back where you began. The same happens with multiplying by 7 and dividing by 7.
Each operation has one partner that cancels it: addition and subtraction form one pair, multiplication and division the other. In an equation, the variable is wrapped in an operation. Applying the partner to both sides peels that layer off and leaves the variable alone. Squaring and square roots are another pair, met in later grades.
Formula and notation
For any numbers \(a\) and \(x\):
\[(x + a) – a = x \qquad (x – a) + a = x\]\[(x \times a) : a = x \qquad (x : a) \times a = x\]
For the last two, \(a\) cannot be 0.
Key facts
- Addition and subtraction are partners; multiplication and division are partners.
- Applying an operation and then its partner returns the original number.
- To solve an equation, use the partner of the operation attached to the variable.
- Undoing must happen on both sides to keep the equation balanced.
Inverse Operations example
A number times 7 equals 91, and a number minus 36 equals 54. Use inverse operations to find both numbers.
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For \(7k = 91\), the variable is multiplied by 7, so divide both sides by 7: \(k = 91 : 7 = 13\).
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For \(m – 36 = 54\), 36 was subtracted, so add 36 to both sides: \(m = 54 + 36 = 90\).
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Check: \(7 \times 13 = 91\) and \(90 – 36 = 54\).
The numbers are 13 and 90.
Common mistake
Watch out
Noel solves \(\frac{y}{6} = 8\) by dividing both sides by 6 and gets \(y = \frac{8}{6}\).
Why it is wrong
The variable is already divided by 6, and dividing again does not undo it. Use the partner: multiply both sides by 6, so \(y = 8 \times 6 = 48\). Check: \(48 : 6 = 8\).
Related terms and lessons
Browse the glossary: A–Z list · Grade 6 vocabulary · Expressions and Equations