# Inverse operations

Here you will learn about inverse operations, including what an inverse operation is and how to use inverse operations to solve one and two-step equations.

Students will first learn about inverse operations as a part of operations and algebraic thinking in 3 rd grade and will expand on their knowledge with negative numbers in 6 th grade.

## What are inverse operations?

Inverse operations are operations which reverse or “undo” another operation. They are also sometimes referred to as ‘opposite operations’.

The four math operations are addition, subtraction, multiplication and division.

Addition and subtraction are inverse operations.

For example,

\begin{aligned}& 4+5=9 \\\\ & 9-4=5 \end{aligned}

Multiplication and division are inverse operations

For example,

\begin{aligned}& 5 \times 8=40 \\\\ & 40 \div 5=8 \end{aligned}

### Solving equations with inverse operations

You can use inverse operations to find the unknown number in one and two-step math equations.

For example,

You can find the unknown number to make the following equation, 56+x=174, true by subtracting.

\begin{aligned}&\begin{aligned}& 56+x=174 \\\\ & 174-56=118 \end{aligned} \\\\ &x=118 \end{aligned}

Note: You can solve inequalities and linear equations using inverse operations similarly to how you can solve equations.

### Inverse operations and negative numbers

Using inverse operations with negative numbers works similarly to using inverse operations with positive numbers. However, you will need to pay special attention to the signs.

Negative numbers are values that are less than zero.

Adding a negative number is the same as subtracting its positive value, and vice versa.

For example,

3+(- \, 5) is the same as 3-5, which equals 2.

Subtracting a negative number is the same as adding its positive value.

For example,

3-(- \, 3) is the same as 3+3, which equals 6.

Multiplying a number by a negative number changes its sign.

For example,

4 \times- \, 5=- \, 20 because multiplying a positive by a negative results in a negative product.

## Common Core State Standards

How does this relate to 3 rd grade and 6 th grade math?

• Grade 3: Operations and Algebraic Thinking (3.OA.A.4)
Determine the unknown whole number in a multiplication or division equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 \, \times \, ? = 48, 5 = \, ? \, \div 3, 6 \times 6 = \, ?.

• Grade 3: Operations and Algebraic Thinking (3.OA.C.7)
Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 \times 5=40, one knows 40 \div 5=8 ) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.

• Grade 6: The Number System (6.NS.C.5)
Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.

• Grade 6: Expressions and Equations (6.EE.B.7)
Solve real-world and mathematical problems by writing and solving equations of the form x+p=q and p x=q for cases in which p, q and x are all nonnegative rational numbers.

## How to find the unknown number using inverse operations

In order to find the unknown number using inverse operations:

1. Identify the operation(s) being applied in the equation.
2. Apply the inverse operation(s) and solve.

## Inverse operations examples

### Example 1: one-step equation

Solve for the variable using inverse operations.

x+5=12

1. Identify the operation(s) being applied in the equation.

The variable has 5 added to it.

2Apply the inverse operation(s) and solve.

The inverse operation of addition is subtraction, so subtract 5 from both sides of the equation.

\begin{aligned} x+5&=12 \\\\ -5 & \; \; -5 \\\\ x&=7 \end{aligned}

### Example 2: one-step equation

Solve for the variable using inverse operations.

4 t=36

Identify the operation(s) being applied in the equation.

Apply the inverse operation(s) and solve.

### Example 3: one-step equation

Solve for the variable using inverse operations.

\cfrac{x}{3}=9

Identify the operation(s) being applied in the equation.

Apply the inverse operation(s) and solve.

### Example 4: two-step equation

Solve for y using inverse operations.

5 x+7=32

Identify the operation(s) being applied in the equation.

Apply the inverse operation(s) and solve.

### Example 5: two-step equation

Solve for x using inverse operations.

3(y-2)=9

Identify the operation(s) being applied in the equation.

Apply the inverse operation(s) and solve.

### Example 6: two-step equation

Solve for x using inverse operations.

4 x-3=13

Identify the operation(s) being applied in the equation.

Apply the inverse operation(s) and solve.

### Teaching tips for inverse operations

• When creating lesson plans, be sure to include visual aids, such as number lines, to help students visualize the inverse operations.

• Provide examples of inverse operations when solving one- and two-step word problems, so that students have examples to refer back to when needed.

• Connect inverse operations to a topic students are already familiar with, like fact families. This will allow them to connect something they know with the new information they are learning.

• Provide students with practice, whether worksheets or other methods, with gradual complexity. This will allow students to become more comfortable with the concept, and feel confident moving to more difficult problems.

### Easy mistakes to make

• Confusing the inverse operations
Students may be confused about what the inverse operation is when first learning. They may think because multiplication and addition are so closely related, that they are inverse operations. Provide students a ‘cheat sheet’ to refer to when needed.

• Forgetting to use the inverse operation on both sides of the equation
When students begin using inverse operations to solve equations, it’s important to reinforce that what is done on one side of the equation must also be done on the other side of the equation.

• Incorrectly following the order of operations
Students may believe that they are to follow the order of operations when applying the inverse operation to an equation. Students should remember that they are to reverse the order of operations as they are solving these equations for variables.

### Practice inverse operations questions

1. Solve for the variable using inverse operations.

b+4=9

b=13

b=36

b=\cfrac{4}{9}

b=5

Apply the inverse of addition to both sides of the equation.

\begin{aligned} b+4 & =9 \\\\ -4 & \;\; -4 \\\\ b & =5 \end{aligned}

2. Solve for the variable using inverse operations.

\cfrac{y}{5}=9

y=\cfrac{9}{5}

y=45

y=14

y=4

Apply the inverse of division to both sides of the equation.

\begin{aligned} \cfrac{y}{5} & =9 \\\\ \times \, 5 & \;\; \times 5 \\\\ y & =45 \end{aligned}

3. Solve for the variable using inverse operations.

m-9=81

m=72

m=9

m=90

m=45

Apply the inverse of subtraction to both sides of the equation.

\begin{aligned} m-9 & =81 \\\\ +9 & \;\; +9 \\\\ m &= 90 \end{aligned}

4. Solve for the variable using inverse operations.

5 y+6=21

y=3

y=6

y=10

y=13

Apply the inverse operations in reverse order, with the inverse of addition (subtraction) first, and then the inverse of multiplication (division) second.

\begin{aligned} 5 y+6 & =21 \\\\ -6 & \;\; -6 \\\\ 5 y &=15 \\\\ \div \, 5 & \;\; \div 5 \\\\ y & =3 \end{aligned}

5. Solve for the variable using inverse operations.

7 z-9=12

z=\cfrac{3}{7}

z=3

z=147

z=21

Apply the inverse operations in reverse order, with the inverse of subtraction (addition) first, and then the inverse of multiplication (division) second.

\begin{aligned} 7 z-9 & =12 \\\\ +9 & \;\; +9 \\\\ 7 z & =21 \\\\ \div \, 7 & \;\; \div 7 \\\\ z & =3 \end{aligned}

6. Solve for the variable using inverse operations.

2 a+8=20

a=6

a=2

a=14

a=18

Apply the inverse operations in reverse order, with the inverse of addition (subtraction) first, and then the inverse of multiplication (division) second.

\begin{aligned} 2 a+8 & =20 \\\\ -8 & \;\; -8 \\\\ 2 a & =12 \\\\ \div \, 2 & \;\; \div 2 \\\\ a & =6 \end{aligned}

## Inverse operations FAQs

What is the inverse function?

The inverse function of a function f is a function that reverses the operation of f.

What is the multiplicative inverse?

The multiplicative inverse of a number is another number which, when multiplied with the original number, yields the product of 1. This is also commonly referred to as the reciprocal.

The additive inverse of a number is another number which, when added to the original number yields a sum of zero. The additive inverse is the negative of the original number.

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