Inverse operations are pairs of operations that undo each other. For example,

subtraction

is the inverse operation of

addition .

When we

add

a

number

and

subtract

the

same

number,

they cancel each other out.

x + 2 - 2 = x + 2 - 2 = x

Division

is the inverse operation of

multiplication

because they undo each other.

y \cdot 3 \div 3 = y

In equations with variables on both sides, we want to gather variable terms on one side of the equation and numbers on the other side. Let's solve an example equation. We will start by combining like terms.

7 x - 3 - 2 x = x - 4 + 9

AddSubTerms

Add and subtract terms

5 x - 3 = x + 5

After we simplify, we use inverse operations to cancel out some terms. On the left-hand side of the equation we will only keep the

x -

terms and on the right-side of the equation we will only keep the numbers.

x - terms = numbers

This means that we want to cancel out

x

on the right-hand side. Let's use inverse operations to do that!

5 x - 3 = x + 5

SubEqn

$LHS - x = RHS - x$

5 x - 3 - x = x + 5 - x

SubTerms

Subtract terms

4 x - 3 = 5

The term disappears on one side of the equation and appears on the other side with the opposite sign. Remember that it is important to apply inverse operations on both sides of the equation! Only then we get an equivalent equation — an equation with the same solutions. Following the same way of thinking, we can solve the equation completely.

4 x - 3 = 5

AddEqn

$LHS + 3 = RHS + 3$

4 x - 3 + 3 = 5 + 3

AddTerms

Add terms

4 x = 8

SplitIntoFactors

Split into factors

x \cdot 4 = 8

DivEqn

$LHS / 4 = RHS / 4$

(x \cdot 4) \div 4 = 8 \div 4

CalcQuot

Calculate quotient

x = 2

In summary, inverse operations allow us to move terms from one side of the equation to the other. This process does not change the solutions of the equation.

Exercise