Lesson 24: Inverse Operations

New Concept

Addition and subtraction are inverse operations. One operation undoes the other.

What’s next

Next, you'll use this concept to rewrite equations and find the missing number in addition and subtraction problems.

Property

Addition and subtraction are inverse operations, which means one operation can undo the other. An addition or subtraction fact can be written in a family of four related equations.

Examples

• The fact family for $8+7=15$ also includes $7+8=15$, $15-7=8$, and $15-8=7$.
• To solve for $n$ in $n+10=25$, you can use the inverse operation: $25-10=n$.
• If you know $30-k=12$, you also know that $12+k=30$.

Explanation

Think of addition and subtraction as a team of superheroes that can reverse each other's actions! If you know one fact, like an addition problem, you can use its inverse partner, subtraction, to find a mystery number. This lets you flip equations around to make them easier to solve, like rearranging puzzle pieces to see the whole picture.

Property

To find a missing addend in an addition equation, you subtract the known addend from the sum. If $a + b = c$, then $c - a = b$.

Examples

• To solve for $m$ in $23 + m = 42$, use subtraction: $m = 42 - 23$, so $m=19$.
• Find the value of $q$ in $q + 17 = 45$. The inverse is $q = 45 - 17$, which means $q = 28$.

Explanation

When a number is hiding in an addition problem, subtraction is your detective! To find the missing piece, just take the total (the sum) and subtract the part you already know. The answer you get is the secret number you were looking for. It's the inverse operation coming to the rescue, showing you exactly how to find the missing value.

Property

To find the starting number in a subtraction problem (the minuend), you add the other two numbers together. If $t - b = c$, then $b + c = t$.

Examples

• To solve for $t$ in $t - 29 = 57$, you can use addition: $t = 57 + 29$, so $t=86$.
• Find the value of $n$ in $n - 26 = 68$. You can find it by adding: $n = 68 + 26$, which means $n = 94$.

Explanation

If you don't know what number you started with in a subtraction problem, don't worry! Just take the amount that was subtracted and add it to the final result. Those two parts will combine to rebuild your original, bigger number. It's like putting two puzzle pieces back together to see the full picture you started with.