Solve linear equations in one variable

We will learn how to solve linear equations using mathematical operations, like addition, subtraction, multiplication and division. 

Here are steps to solve a linear equation in one variable.

When variables on both sides of the equations, then following steps are followed for solving the equations.

Some examples are given to solving linear equations in one variable.

           Examples for solving linear equations 

1. Solve the equation  3x + 4 = 5x – 2

Solution:

Step 1: To solve this equation first, we arrange the variables in one side and the numbers are in other side.

First we transfer +3x from left hand side to right hand side, so positive 3x change in to negative -3x and equation becomes

                       4 = 5x – 2 – 3x

Step 2: Similarly transpose all the numbers on the other side of the equation. 

again transfer -2 from right hand side to left hand side so negative 2 change in to positive 2

equation will be    4 + 2 = 5x – 3x

                  or            6 = 2x

Step 3: Divide the equation with 2 on both sides of equality

                   divide both sides by 2

                  or         6/2 = 2x/2
                  or            3 = x
                  or            x = 3

 If we substitute  the value of x is 3 in the equation we will get 3 = 3, so 3 is the solution of the given equation.

            The solution of the equation is 3

2. Solve the equation 6x = 3x + 1/2

Solution: The equation is 6x = 3x + 1/2

Transfer 3x from right hand side to left hand side, then positive 3x change in to negative 3x.

                             6x – 3x = 1/2

                                    3x = 1/2

                         divide both sides by 3

                                 3x/3 = 1/2 x 1/3

                                      x = 1/6

    Therefore 1/6 is solution of the  given equation.

3.  Solve the equation   3(x – 3) = 5(2x + 1)

Solution: In L.H.S. multiply 3 to (x – 3) and R.H.S. multiply   5 to (2x + 1).

3x – 9 = 10x + 5
                     3x – 10x = 5 + 9
                            -7x = 14
                         -7x/7 = 14/7
                               -x = 2
                                x = -2

    Therefore -2 is solution of the  given equation.

4.  Solve the equation 5x + 7/2 = 3x/2 – 14

Solution: multiply both sides of equation by 2, we get

2 x (5x + 7/2) = 2 x (3x/2 – 14)

(2 x 5x) + (2 x 7/2) = (2 x 3x/2) – (2 x 14)

  or                 10x + 7 = 3x – 28

 or              10x – 3x + 7 = – 28  (transposing 3x, RHS to LHS)

              7x + 7 = – 28

7x = – 28 – 7 

                                7x = – 35

                       x = – 35/7

                          x = – 5  (solution)

5.  Solve the equation  2x/3 + 1 =7x/15 + 3

Solution: transposing 7x/15 to LHS and 1 to RHS

2x/3 – 7x/15 = 3 – 1

LCM of 3 and 15 is 15.

or     (10x – 7x)/15 = 2

                or             (3x)/15 = 2

or                    3x = 2 x 15

     or                    3x = 30

or         x = 30/3

or               x = 10 (solution)