Equations are mathematical statements which occupy a central place in solving problems related to any topic in Math. An equation is a mathematical statement in which two mathematical expressions are related by an equal sign (=). Solving an equation means finding out what number x stands for, if x is a variable.

An equation is a mathematical statement that tells two algebraic expressions are equal.

**Definition of Equation:**An equation is a mathematical statement that tells two algebraic expressions are equal.

## Solution of an Equation

Example: x + 6 = 10, the value of x only make the equation true.

x = 4, because 4 + 6 = 10.

Equation with two variable instead of one. Solution to the equation x + y = 1 will be not a single number but a pair of numbers, one for x and one for y, that makes the equation true.

## Properties of an Equality

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1

**. The addition property of equality**-

It states that if a = b, then a + c = b + c. The quantity c was added to both sides so that the balance is maintained.

Example: Value of x, x + 1 = 3**Case 1:** Solution of x + 1 = 3

x = 3 - 1

= 2

**Case 2:** Adding 4 both side, we have,

x + 1 + 4 = 3 + 4

x + 5 = 7

x = 7 - 5

= 2

So, in both the case the value of x is 2.

**2. The multiplication property of equality:**

It states that if a = b, then a * c = b * c.

**The quantity c multiplied both sides so that the balance is maintained.**

Example: Value of x, x * 2 = 4

then x = 2

if we multiply 3 both side,

then x * 2 * 3 = 4 * 3

x = 2.

** 3. The substitution property of equality:**

The substitution property is same as addition property of equality. It states that if a = b, then a - c = b - c. The quantity c was added to both sides so that the balance is maintained**.**

## How to Solve an Equation

**Steps For Solving the equation:**

**Step 1:**Use the distributive property to separate terms, if necessary.

**Step 2:**Use addition or subtraction property of equality to get all variable terms on one side of the equation and all constant terms on the other side.

**Step 3:**Use the multiplicative property of equality to get value of the unknown.

**Example:**

Solve x + 7 = 15

Using the subtraction property of equality, subtracting 7 from both side of the equation will resulting in altering the given equation.

x + 7 - 7 = 15 -7

x = 8

## Transposition Rules

- If a number is added to a variable or to an expression containing the variable, then subtract the number on the other side.
- If a number is subtracted to a variable or to an expression containing the variable, then add the number on the other side.
- If a variable or an expression containing a variable is multiplied by a number, then divide the expression on the other side by the number.
- If a variable or an expression containing a variable is divided by a number, then multiply the expression on the other side by the number.

## Solving One Step Equations

**One step equation,**an equation that can be solved in one step by using transposition rules.

### Solved Examples

**Question 1:**Find the value of x, x + 8 = 18

**Solution:**

x + 8 = 18

Subtracting 8 from both side, we have

x + 8 - 8 = 18 - 8

x = 10

Subtracting 8 from both side, we have

x + 8 - 8 = 18 - 8

x = 10

**Question 2:**Find the value of y, y - 3 = 12

**Solution:**

y - 3 = 12

adding 3 from both side, we have

y - 3 + 3 = 12 + 3

y = 15

adding 3 from both side, we have

y - 3 + 3 = 12 + 3

y = 15

**Question 3:**Find the value of x, 3x = 24

**Solution:**

3x = 24

Multiplying by $\frac{1}{3}$ from side, we have

3x * $\frac{1}{3}$ = 24 * $\frac{1}{3}$

x = 8.

Multiplying by $\frac{1}{3}$ from side, we have

3x * $\frac{1}{3}$ = 24 * $\frac{1}{3}$

x = 8.

## Solving Literal Equations

### Solved Examples

**Question 1:**Solve this literal equation for x, 2x + 5y = x + 2y

**Solution:**

2x + 5y = x + 2y

On subtracting x from both side,

2x + 5y - x = x + 2y - x

(2x - x) + 5y = (x - x) + 2y

x + 5y = 2y

On subtracting 5y from both sides,

x + 5y - 5y = 2y - 5y

x = (2 - 5)y

x = - 3y.

On subtracting x from both side,

2x + 5y - x = x + 2y - x

(2x - x) + 5y = (x - x) + 2y

x + 5y = 2y

On subtracting 5y from both sides,

x + 5y - 5y = 2y - 5y

x = (2 - 5)y

x = - 3y.

**Question 2:**Solve this literal equation for p, x + 2p = 5x - 4p

**Solution:**

x + 2p = 5x - 4p

On subtracting x from both side

x + 2p - x = 5x - 4p - x

2p = (5x - x) - 4p

2p = 4x - 4p

Adding 4p from both side,

2p + 4p = 4x - 4p + 4p

(2 + 4)p = 4x

6p = 4x

By dividing 6, we have

$\frac{6p}{6}$ = $\frac{4x}{6}$

p = $\frac{2x}{3}$

On subtracting x from both side

x + 2p - x = 5x - 4p - x

2p = (5x - x) - 4p

2p = 4x - 4p

Adding 4p from both side,

2p + 4p = 4x - 4p + 4p

(2 + 4)p = 4x

6p = 4x

By dividing 6, we have

$\frac{6p}{6}$ = $\frac{4x}{6}$

p = $\frac{2x}{3}$