Inverse Operations
In short: Inverse operations come in matched pairs where the second reverses the first, so plus 15 is canceled by minus 15 and times 4 by divide by 4.
6th GradeAlgebra6.EE.B.7
What is Inverse Operations?
Inverse operations work like putting on a coat and taking it off: you end up where you started. Adding 15 and then subtracting 15 returns the original value, and so does multiplying by 4 and then dividing by 4.
Precisely, for any number \(n\): \((n + a) – a = n\), \((n – a) + a = n\), \(\frac{n \cdot a}{a} = n\) and \(\frac{n}{a} \cdot a = n\) when \(a \neq 0\). Squaring and taking a square root also undo each other for nonnegative numbers.
How to spot it
- A variable is stuck to a number by adding, subtracting, multiplying or dividing and must be freed.
- A calculation is checked by working backward, such as testing a division with multiplication.
- A story asks what you started with after a sequence of changes.
- The words “undo” or “opposite operation” are used.
The method
- Identify the operation that is acting on the variable or the number.
- Choose its partner: plus pairs with minus, times pairs with divide.
- Apply the partner to both sides of an equation, using the same number.
- Verify the answer by running the original operation forward.
Inverse Operations: solved example
Problem
A hiker in Montana buys a pack that was marked down by $18 and now costs $67. Use inverse operations to find the original price.
Solution
Write \(p – 18 = 67\). The operation on p is subtracting 18, so the partner is adding 18 to both sides.
\(p = 67 + 18 = 85\). Check: \(85 – 18 = 67\). The original price was $85.
Try it
Question 1
Which operation undoes “divide by 12”?
Show the answer
Multiplying by 12.
Question 2
A number is tripled and the result is 51. Work backward to find the number.
Show the answer
Undo times 3 by dividing: \(51 \div 3 = 17\).
Inverse Operations vs. Order of Operations
The order of operations decides which step to do first when evaluating a line forward. Inverse operations are used going backward, to undo a step, and when you solve an equation you usually undo the last-done operation first.
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Whole dictionary: A–Z · 6th Grade · Algebra