Let us first understand the meaning of inverse operation.

**What is Inverse Operation?**

Any **math calculation which have opposite affect in the calculation** is called inverse operation.

For Example, Addition and Subtraction have opposite affect in the calculation.

Similarly multiplication and division have opposite effect, hence they are considered as inverse operation in mathematics.

**Division and Multiplication as Inverse operation**

Division and multiplication have opposite affect in number calculation.

To understand the concept, let us consider a number 5.

Multiply number 5 with 7, you will get number 35;

⟹ 5 x 7 = 35

Now, if you want to get number 5 back, divide number 35 by 7;

⟹ 35 ÷ 7 = 5

Here we have used division to get back to the original number.

Hence, division works opposite to multiplication.

**How this inverse property helpful for students?**

The property is helpful to solve complex algebra problems in fast and easy way.

If the given equation is: **A ÷ B = C**

Then using the above property, the equation can be written as:**A = B x C**

**Example:**

(a) 6 ÷ 2 = 3

We can also write; 3 x 2 = 6

(b) 45 ÷ 5 = 9

Or, 9 x 5 = 45

(c) 72 ÷ 12 = 6

Or, 12 x 6 = 72

(d) 96 ÷ 8 = 12

Or, 12 x 8 = 96

(e) 75 ÷ 15 = 5

Or, 15 x 5 = 75

**Using inverse property to solve Algebraic Equation**

In algebraic equation, **moving from one side to another side the division will change into multiplication or vice versa**.

Observe the below image.

Note how the **division is changed into multiplication** in algebraic equation.

Similarly the **multiplication can be changed into division **on moving to other side of the algebra equation.

Let us see some examples to fully understand the concept.

**Example 01**

Find the value of x in given equation

\mathtt{\frac{x}{2} \ =\ 7}\\\ \\

\mathtt{The\ equation\ can\ be\ written\ as:}\\\ \\ \mathtt{x\ =\ 7\ \times \ 2}\\\ \\ \mathtt{x\ =\ 14} \\\ \\

Hence x = 14 is the solution.

**Example 02**

Find the value of y in below equation

\mathtt{\frac{8y}{3} \ =\ 16} \\\ \\

\mathtt{The\ equation\ can\ be\ written\ as:}\\\ \\ \mathtt{8y\ =\ 16\ \times \ 3}\\\ \\ \mathtt{8y\ =\ 48}\\\ \\ \mathtt{Move\ 8\ towards\ right}\\\ \\ \mathtt{y\ =\ \frac{48}{8}}\\\ \\ \mathtt{y\ =\ 6}

**If inverse of multiplication is division then what is the inverse of subtraction?**

Addition and Subtraction are inverse to each other.

While addition add numbers, subtraction reduce it.

Suppose we have number 5 and we add 2 in it.

5 + 2 = 7

Now in order to get 5 back, subtract 2 from 7

7 – 2 = 5

Hence, both addition and subtraction works in opposite way.

**Solved Problems – Division as Inverse of Multiplication**

**(01) Find the value of x from below equation** \mathtt{\frac{2x\ +\ 8}{4} \ =\ 7} \\\ \\

\mathtt{Move\ 4\ to\ the\ right}\ \\\ \\ \mathtt{2x\ +\ 8\ =\ 7\ \times \ 4}\ \\\ \\ \mathtt{2x\ +\ 8\ =\ 28}\ \\\ \\ \mathtt{Move\ 8\ to\ the\ right}\ \\\ \\ \mathtt{2x\ =\ 28\ -\ 8}\ \\\ \\ \mathtt{2x\ =\ 20}\ \\\ \\ \mathtt{Move\ 2\ to\ the\ right}\ \\\ \\ \mathtt{x\ =\ \frac{20}{2}}\ \\\ \\ \mathtt{x\ =\ 10 } \\\ \\

**Hence, the value of x is 10.**

**(02) Find the value of y in given equation**

\mathtt{\frac{3y\ }{11} \ =\ 5y\ -\ 1} \\\ \\

\mathtt{Moving\ 11\ to\ the\ right.}\ \\ \\ \mathtt{Division\ will\ turn\ into\ multiplication}\ \\\ \\ \mathtt{3y\ =\ 11\ \times \ ( 5y\ -1)}\\\ \\ \mathtt{3y\ =\ 55y\ -\ 11}\ \\\ \\ \mathtt{Moving\ 11\ to\ the\ left}\ \\ \\ \mathtt{Subtraction\ will\ become\ addition}\ \\\ \\ \mathtt{3y\ +\ 11\ =\ 55y}\ \\\ \\ \mathtt{Moving\ 3y\ towards\ right}\ \\ \\ \mathtt{Addition\ will\ become\ subtraction}\ \\\ \\ \mathtt{11\ =\ 55y\ -\ 3y\ }\ \\\ \\ \mathtt{11\ =\ 52y}\ \\\ \\ \mathtt{y\ =\ \frac{11}{52}} \\\ \\

**Hence, the value of y is 11/52**

**(03) Find the value of y in given equation**

\mathtt{y-2\ =\ \frac{3y}{7}} \\\ \\

\mathtt{Move\ 7\ to\ the\ left}\\\ \\ \mathtt{Division\ will\ turn\ into\ multiplication}\ \\\ \\ \mathtt{7\ ( y\ -\ 2) \ =\ 3y}\ \\\ \\ \mathtt{7y\ -\ 14\ =\ 3y}\ \\\ \\ \mathtt{Move\ 3y\ towards\ left}\ \\ \\ \mathtt{Addition\ becomes\ subtraction}\ \\\ \\ \mathtt{7y\ -\ 3y\ -\ 14\ =\ 0}\ \\\ \\ \mathtt{4y\ -\ 14\ =\ 0}\ \\\ \\ \mathtt{Move\ 14\ towards\ right}\ \\ \\ \mathtt{Subtraction\ become\ addition}\ \\\ \\ \mathtt{4y\ =\ 14} \\\ \\ \mathtt{y\ =\ \frac{14}{4}} \\\ \\

**Hence, the value of y is 14/4**