Substitution method 
for Solving linear Systems

The Substitution method for Solving linear Systems is easy to solve the equations.

Substitution method can be applied in below steps.

Step 1: First select one equation and solve it for one variable either x = or y =.

Step 2: Substitute the solution from step 1 into the other equation.

Step 3: Solve the new equation.

Step 4: Substitute the value in equation and solve it for second variable.

Step 5: Check the solutions in both original equations.

Solve the following pair of linear equations by substitution method.

Example: (1)     x + y = 14
                        x – y  = 4

Solution:  let     x + y = 14……(1)
                        x – y  = 4…….(2)

Step 1: Solve one of the equations for either x = or y =.

We will solve first equation for x

                         x + y = 14…..(1)

                         x = 14 – y 

Step 2: Substitute the solution from step 1 into the other equation, that is equation 2.

                            x – y = 4……(2)

                        (14 – y) – y = 4

Step 3: Solve the new equation.

                         14 – y – y = 4

                            14 – 2y = 4

                                  -2y = 4 – 14

                                  -2y = -10

                                     y = -10/-2

                                     y = 5

Step 4: Substitute the value in equation and solve it for second variable.

                             x + y = 14

                             x + 5 = 14

                                    x = 14 – 5

                                    x = 9

Step 5: Check the solutions in both original 
equations.

   x + y = 14…..(1)              x – y = 4…..(2)

  9 + 5 = 14                       9 – 5 = 4

      14 = 14                            4 = 4

           The solution is (x, y) = (9, 5)

Example:(2)     2x + 5y = 12
                        4x – y  = 2

Solution:  let     2x + 5y = 12……(1)
                        4x – y  = 2…….(2)

Step 1- Solve one of the equations for either x = or y =.

We will solve second equation for y

                         4x – y = 2…..(1)

                         -y = 2 – 4x

                          y = 4x – 2 

Step 2- Substitute the solution from step 1 into the other equation, that is equation 1.

                          2x + 5y = 12……(1)

                   2x + 5(4x – 2) = 12

Step 3- solve the new equation.

                      2x + 20x – 10 = 12

                            22x = 12 = 10

                                 22x = 22

                                  x = 22/22

                                     x = 1/1

                                     x = 1

Step 4- Substitute the value in equation and solve it for second variable.

                             2x + 5y = 12

                             2 x 1 + 5y = 12

                                   2 + 5y = 12

                                         5y = 12 – 2

                                         5y = 10

                                          y = 10/5

                                          y = 2

Step 5- Check the solutions in both original 
equations.

  2x + 5y = 12….(1)                4x – y = 2….(2)

  2 x 1 + 5 x 2 = 12                 4 x 1 – 2 = 2

            2 + 10 = 12                    4 = 4

           The solution is (x, y) = (1, 2)

Example:(3)     2x – 9y = 0
                        x – 18y  = 27

Solution:  let     2x – 9y = 0…….(1)
                       x – 18y  = 27…….(2)

Step 1- Solve one of the equations for either x = or y =.

We will solve second equation for x

                         x – 18y = 27…..(1)

                         x = 27 + 18y 

Step 2- Substitute the solution from step 1 into the other equation, that is equation 1.

                        2(27 + 18y) – 9y = 0

Step 3- solve the new equation.

                         54 + 36y – 9y = 0

                            27y = – 54

                                  y = – 54/27

                                  y = – 2

                                  y = – 2

Step 4- Substitute the value in equation and solve it for second variable.

                           2x – 9y = 0

                           2x – 9(-2) = 0

                              2x + 18 = 0

                                    2x = -18

                                      x = -18/2

                                       x = -9

Step 5- Check the solutions in both original 
equations.

2x – 9y = 0…..(1)              x – 18y = 27…..(2)

2 x (-9) – 9 x (-2)= 0        -9 – 18 x (-2) = 27

-18 +18 = 0                       -9 + 36 = 27

      0 = 0                    27 = 27

           The solution is (x, y) = (-9, -2)