## What is Addition Property of Equality?

The property says that if we add numbers on both side of the balanced equation, the equation will still be balanced and valid.

Example
Let the given equation is Ax + By = C

If we add number D on both sides, the equation will still be balanced

Ax + By + D = C + D

Generally the addition property of equality is expressed as:

Where A, B & x are real numbers

### Verification of addition property of equality

Consider the below equation:

5 = 5 – – -(i)

Above equation is true & balanced.

Add number 6 on both sides of equation

5 + 6 = 5 + 6

11 = 11 – – -(ii)

Even after adding numbers, the equation (ii) is still true.

Hence, doing math operation on both side of equation does not affect equation quality.

### How addition property of equality works

Suppose you have been provided with following balanced equation

2x + 3 = 20

Imagine the equation as the below see-saw whose right and left weight are completely balanced

Now if you add any number on both side the equation will still be balanced.

Suppose you add number 10 on both sides. In this case, the see-saw is still balanced.
2x + 3 + 10 = 20 + 10

But if you add number only on one side, the balance get disturbed & the equation may not hold valid

How is the property useful?

In your higher mathematics class you will come across complex algebraic and calculus equation which will take lot of your time and energy.

On using this property you will be able to solve equation in fast and easy steps.

## Addition Property of equality examples

Example 01
Solve the equation and find value of x
x – 8 = 12

Add 8 on both sides so that only variable x remains on left side

x – 8 + 8 = 12 + 8

x = 20

Hence, the value of x is 20

Example 02
Solve the equation
5x – 15 = 65

Add 15 on both sides so that only variable x remain on left side
5x – 15 + 15 = 65 + 15

5x = 80

x = 16

Hence, the value of x is 16

Multiple Variables on one side

Example 01
11x – 14 – 3x + 4 = 7

11x – 3x – 14 + 4 = 7

8x – 10 = 7

Add 10 on both sides, so that only variable x remain on left

8x – 10 + 10 = 7 + 10

8x = 17

x = 17/8

The value of x is 17/8

Example 02
11 = 25 + 9x – 100 + 6x

11 = 25 – 100 + 9x + 6x

11 = -75 + 15x

Add 75 on both sides so that only variable x remains on right

11 + 75 = -75 + 75 + 15x

86 = 15x

x= 86/15

The value of x is 86/15

Variable on both side of equation

Example 01
6x = -3x + 16

Add 3x on both sides to remove variable x from right side

6x + 3x = -3x + 3x + 16

9x = 16

x = 16/9

The value of x is 16/9

Example 02
10x + 2 = 16x

Subtract 10x on both sides to remove variable x from left

10x – 10x + 2 = 16x – 10x

2 = 6x

x =2/6

The value of x is 2/6

Multiple variable on both sides

Example 01
16x + 9 – 21x = 3x + 5

16x – 21x + 9 = 3x + 5

-5x + 9 = 3x + 5

-5x + 5x + 9 = 3x + 5x + 5

9 = 8x + 5

Subtract 5 on both sides

9 – 5 = 8x + 5 -5

4 = 8x

x = 4/8 = 1/2

Hence, the value of x is (1/2)

Example 02
12x + 33 + 13x = 17x – 63 – 45x

12x + 13x + 33 = 17x – 45x – 63

25x + 33 = -28x -63

25x + 28x + 33 = – 28x + 28x – 63

53x + 33 = – 63

Subtract 33 on both sides

53x + 33 – 33 = – 63 – 33

53x = -99

x = -99/53

Hence, the value of x is -99/53

(01) Will the property of equality work for subtraction also?

Yes!!

There is a subtraction property of equality which says that if you subtract same number on both side of balanced equation, the equation will remain still valid.

if Ax + B = C is the given equation

Subtract number D on both sides of equation

Ax + B – D = C – D

This equation is valid and balanced

(02) How Commutative property and Addition Property of equality different?

Commutative property says that in addition, the change in the order of number will have no effect on the final result.

A + B = B + A

A = B

A + C = B + C

(03) Will the equality property works for multiplication?

Yes!!

In a balanced equation, if you multiply a number on both side of the equation, it will still remain valid and balanced.

Let the given equation be: Ax + B = C

Multiply number D on both sides, we get:

D (Ax + B) = C. D