# Multiplication property of equality

## Multiplication property of equality definition

The property states that in a balanced algebraic equation, if we multiply a number on both sides the equation will remain still balanced and valid.

Example
Let the given equation is Ax + By = C

If we add number D on both side, the equation becomes:
D . (Ax + By) = D. C

A. D. x + B. D. y = D. C

According to the property, the new equation is valid and balanced.

Generally the multiplication property of equality is expressed as:

Where A, B & x can be any possible real numbers.

### Verification of Multiplication property of Equality

The below equation is balanced.

11 = 11 – – -(i)

Now multiply both side with number 5

5 x 11 = 5 x 11

55 = 55 – – -(ii)

Observe that even after multiplication, the equation is still equal and balanced.
Hence, multiplication property of equality is verified.

### How multiplication property of equality works

Suppose the below given equation is balanced.
7x + 6 = 48

Imagine a balanced equation as see-saw where both left & right sides are balanced.

Now if you multiply both sides of the equation with random number, the equation will still be balanced.

Let’s multiply the above equation with 2.

2 (7x + 6 ) = 48 . 2

14 x + 12 = 96

The equation is still balanced.
Given below is the see – saw representation of above equation.

However, if you multiply the equation only on one side or multiply the sides with different number, the equation will get imbalanced.

Let’s multiply one side of the equation by 3
7x + 6 = 48
3 (7x + 6 ) and 48
21 x + 18 and 48

The equation gets unbalanced.
Given below is the representation of unbalanced equation in the form of see-saw.

### Multiplication property of Equality Example

Example 01
\mathtt{\ \frac{x}{2} \ =\ 16}\\\ \\ \mathtt{Multiply\ 2\ on\ both\ sides\ }\\\ \\ \mathtt{2\ \times \frac{x}{2} \ =\ 16\times 2}\\\ \\ \mathtt{\frac{2x}{2} \ =\ 32}\\\ \\ \mathtt{x\ =\ 32}\\\ \\ \mathtt{Hence,\ value\ of\ x\ is\ 32}

Example 02
\mathtt{\frac{x+6}{3} \ =\ 21}\\\ \\ \mathtt{Multiply\ 3\ on\ both\ sides\ }\\\ \\ \mathtt{3\ \times \frac{x+6}{3} \ =\ 21\ \times \ 3}\\\ \\ \mathtt{\frac{3\ ( x+6)}{3} \ =\ 63}\\\ \\ \mathtt{x\ +\ 6\ =\ 63}\\\ \\ \mathtt{subtract\ 6\ on\ both\ sides}\\\ \\ \mathtt{x\ +\ 6\ -\ 6\ =\ 63\ -\ 6}\\\ \\ \mathtt{x\ =\ 57}\\\ \\ \mathtt{Hence,\ value\ of\ x\ is\ 57}\

Multiple Variable on one side

Example 01
\mathtt{\frac{10x\ +\ 10\ +\ 2x}{5} =\ 15}

\mathtt{\frac{10x\ +\ 2x+\ 10}{5} =\ 15}\\\ \\ \mathtt{\frac{12x\ +\ 10}{5} \ =\ 15}\\\ \\ \mathtt{Multiply\ both\ sides\ by\ 5}\\\ \\ \mathtt{\frac{5\ ( 12x\ +\ 10)}{5} \ =\ 15\ \times \ 5}\\\ \\ \mathtt{12x\ +\ 10\ =\ 75}\\\ \\ \mathtt{subtract\ 10\ from\ both\ sides}\\\ \\ \mathtt{12x\ +\ 10\ -\ 10\ =\ 75\ -\ 10}\\\ \\ \mathtt{12x\ =\ 65}\\\ \\ \mathtt{x\ =\ \frac{65}{12}}

Example 02
\mathtt{\frac{13}{7} \ =\ 6y\ +\ 3\ +\ 4y}

\mathtt{\frac{13}{7} \ =\ 6y\ +\ 4y\ +\ 3}\\\ \\ \mathtt{\frac{13}{7} \ =\ 10y\ +\ 3}\\\ \\ \mathtt{Multiply\ both\ sides\ by\ 7}\\\ \\ \mathtt{7\left(\frac{13}{7}\right) \ =\ 7.\ ( 10y\ +\ 3)}\\\ \\ \mathtt{Apply\ distributive\ property}\\\ \\ \mathtt{a\ ( b+c) \ =\ ab\ +\ ac}\\\ \\ \mathtt{13\ =\ 7.\ 10y\ +\ 7.3}\\\ \\ \mathtt{13\ =\ 70y\ +\ 21}\\\ \\ \mathtt{Subtract\ 21\ on\ both\ sides}\\\ \\ \mathtt{13\ -\ 21\ =\ 70y\ +\ 21\ -\ 21}\\\ \\ \mathtt{-8\ =\ 70y}\\\ \\ \mathtt{y\ =\ \frac{-8}{70} \ }

Variables on both sides

Example 01
\mathtt{\frac{7y}{4} \ =\ 11y\ +\ 21\ -\ 3y}\\\ \\

\mathtt{\frac{7y}{4} \ =\ 11y\ -\ 3y\ +\ 21}\\\ \\ \mathtt{\frac{7y}{4} \ =\ 8y\ +\ 21\ }\\\ \\ \mathtt{Multiply\ both\ sides\ by\ 4}\\\ \\ \mathtt{4\ \left(\frac{7y}{4}\right) \ =\ 4\ ( 8y\ +\ 21)}\\\ \\ \mathtt{7y\ =\ 32y\ +\ 84}\\\ \\ \mathtt{Subtract\ both\ sides\ by\ 32y}\\\ \\ \mathtt{7y\ -\ 32\ y\ =\ 32y-32y\ +\ 84}\\\ \\ \mathtt{-25y\ =\ 84}\\\ \\ \mathtt{y\ =\ \frac{-84}{25} \ }

Example 02
\mathtt{\frac{6x\ +\ 10}{3} \ =\ \frac{\mathtt{12x\ -\ 6\ -\ } 2x}{2}} \\\ \\

\mathtt{\frac{6x\ +\ 10}{3} \ =\ \frac{\mathtt{12x\ -\ } 2x-6}{2}}\\\ \\ \mathtt{\frac{6x\ +\ 10}{3} \ =\ \frac{\mathtt{10x\ -\ 6}}{2}}\\\ \\ \mathtt{Multiply\ both\ side\ of\ equation\ by\ 3}\\\ \\ \mathtt{\frac{3\ ( 6x\ +\ 10)}{3} \ =\ \frac{3\ (\mathtt{10x\ -\ 6})}{2}}\\\ \\ \mathtt{( 6x\ +\ 10) \ =\ \frac{30x\ -\ 18}{2}}\\\ \\ \mathtt{Multiply\ both\ sides\ by\ 2}\\\ \\ \mathtt{2.\ ( 6x\ +\ 10) \ =\ \frac{2\ ( 30x\ -\ 18)}{2}}\\\ \\ \mathtt{12x\ +\ 20\ =\ 30x\ -\ 18}\\\ \\ \mathtt{Subtract\ both\ sides\ by\ 12x}\\\ \\ \mathtt{12x\ -\ 12x\ +\ 20\ =\ 30x-12x-18}\\\ \\ \mathtt{20=\ 18x-18}\\\ \\ \mathtt{Add\ 18\ both\ sides}\\\ \\ \mathtt{20\ +\ 18\ =18x-18+18}\\\ \\ \mathtt{38\ =\ 18x}\\\ \\ \mathtt{x\ =\ \frac{38}{18}}

### Frequently asked Questions : Multiplication Property of Equality

#### Will the property of equality work for addition and subtraction also?

Yes!!

In a balanced equation, if you add/subtract any number on both side, the equation will remain balanced and valid.

For Example
Consider the below equation as balanced
2x + 3y + 4 = 16

Subtract 2 from both sides
2x + 3y + 4 – 2 = 16 – 2

2x + 3y + 2 = 14
The equation is balanced and valid.

Now add 6 on both sides
2x + 3y + 2 + 6 = 14 + 6
2x + 3y + 8 = 20
The equation is still considered balanced.

#### How multiplication property of equality is different from distributive property?

Distributive property is all about reducing the complexity of equation containing both multiplication and addition
A . (B + C) = A.B + A.C

On the other hand equality property is all about multiplication of balanced equation on both sides.

#### Will the equality property works in division also?

Yes!!
In a given balanced equation, if you divide a number on both sides, the equation will still remain valid.

Consider the equation : 6y = 7x

Divide the equation by 3 on both sides

\mathtt{\frac{6y}{3} =\frac{7x}{3}}

The equation is still valid and balanced.

#### Why this property is called Multiplication Property of Equality?

Because on multiplying the equal equation on both sides, the equation still remains equal.

#### If we multiply the balanced equation with different numbers, will the equation still remain balanced?

No!!
Multiplying the equation with different numbers will disturb the balance of equation and it will no longer remain balanced.

For Example, consider the below balanced equation:
30 = 30
Both left & right side are equal.

Now multiply left side by 2 and right side by 3;
30 x 2 & 30 x 3
60 & 90

Hence, multiplication with different numbers make the equation unbalanced.

#### Will the property work in inequality equation?

Yes!!
Multiplication of same positive number on both sides will not disturb the inequality equation.

Consider the below equation:
(2x + 3y) > 34

Multiply 3 on both sides;
3 (2x + 3y) > 34. 3
6x + 9y > 102

Note: Multiplication of positive number will not affect the inequality equation.

But what happen if we multiply negative number?
Multiplication of negative number will change the inequality sign.

Let us again take the above mentioned example:
(2x + 3y) > 34

Multiply -3 on both sides
(-3) (2x + 3y) < 34. (-3)
-6x – 9y < -102

Note the sign is changed from “>” to “<“

Conclusion
(a) Multiplication with +ve number will not affect inequality equation.
(b) Multiplication with -ve number will change the sign of inequality equation.

## Multiplication property of equality – Solved Problems

(01) Solve the equation and find value of x
\mathtt{\frac{x\ +\ 3}{2} \ =\ 7}

(a) 5
(b) 10
(c) 11
(d) 13

Option (c) is the right answer

Explanation:
\mathtt{\frac{x\ +\ 3}{2} \ =\ 7}\\\ \\ \mathtt{Multiply\ both\ the\ sides\ by\ 2}\\\ \\ \mathtt{\frac{2\ ( x\ +\ 3)}{2} \ =\ 7.2}\\\ \\ \mathtt{x\ +\ 3\ =\ 14}\\\ \\ \mathtt{Subtract\ both\ sides\ by\ 3}\\\ \\ \mathtt{x\ +\ 3\ -\ 3\ =\ 14\ -\ 3}\\\ \\ \mathtt{x\ =\ 11}

(02) Property of equality works in following Math Operation.

(b) Multiplication
(c) Subtraction
(d) Division
(e) All of above

Option (e) is correct

Explanation
Let the balanced equation is: y + z = 3

Adding c on both side of equation.
y + z + c = 3 + c

(b) Subtraction property of equality
Subtract c on both side of equation.
y + z – c = 3 – c

(c) Multiplication property of equality
Multiply c on both side of equation.
c ( y + z ) = 3.c

(d) Division Property of equality
Divide c on both sides
\mathtt{\frac{y\ +\ z\ }{c} =\ \frac{3}{c}}

(03) Solve the below equation and find value of x
\mathtt{\frac{50\ y\ }{2} \ =\ 6y\ +\ 1}

(a) 2/38
(b) 2/39
(c) 2/26
(d) 2/13