Solution to a given equation is the value of the variable, which satisfy the equations.

**Rules to solve equations with variables on both sides:**

**Linear Equations: The general form of linear equations will be ax + b = 0, where a $\ne$ 0. **

In the case of linear equations which have only one variable for which the variables are on both sides, we need to bring the variables to one side of the equal to sign, and the constants to other side of the equation and then eliminate the constants with the variable and solve for x.

### Example: 2x - 5 = 13 - 4x

**Solution: **We have 2x - 5 = 13 - 4x

Eliminating the constant in the left hand side, we get,

2x - 5 + 5 = 13 - 4x + 5

=> 2x = 18 - 4x

=> 2x + 4x = 18 - 4x + 4x

=> 6x = 18

=> x = $\frac{18}{6}$

= 3

Therefore, solution **x = 3**

**Quadratic Equations: The quadratic equations are of the form, ax**^{2} + b x + c = 0, where a $\ne$ 0.

A quadratic equation is a polynomial of degree two. There will be two solutions for the quadratic equations. The solutions are called as the roots of the equations.

The equations can be solved by

a. Factorization

b. Formula Method

c. Completing the square method.

When the equations have variables on both sides, we should follow the rules of equality, such that all the terms containing the variables and the constants are brought to the left side of the equation, so that the final equation is of the form, ax^{2} + bx + c = 0, then we solve the equation by one of the three methods shown above.

### Example: Solve( x + 2 ) = $\frac{(6x+3)}{(x-2)}$

Solution: We have ( x + 2 ) = $\frac{(6x+3)}{(x-2)}$

( x + 2 ) ( x - 2 ) = 6x + 3 [ by multiplying by ( x - 2 ) on both sides ]

x^{2} - 4 = 6x + 3

=> x^{2} - 6x - 4 - 3 = 0

=> x^{2} - 6x - 7 = 0

=> x^{2} - 7x + x - 7 = 0

=> x ( x - 7 ) + 1 ( x - 7 ) = 0

=> ( x - 7 ) ( x + 1 ) = 0

=> x - 7 = 0 or x + 1 = 0

=> x = 7 or x = -1

Therefore **the solution to the above equation is x = { -1, 7}**.