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Solving a System of Two Linear Equations

The system of linear equations is defined as the set of equations in which there are two or more than two equations given and one has to solve for the respective variables.

The graph of the linear equation is a straight line and the value of the variable or solution of the equation is the intersection point on the graph.

There are various ways by which one can solve a system of equations given as below-

  1. Elimination method
  2. Substitution method
  3. Augmented matrix

Let’s discuss each of the methods one by one.

  1. Elimination method

The elimination method is the method in which we eliminate either x or y in order to get the value for the other variable.

Example 1: Solve 2x – 2y = 8

                              x + y = 15

Now, as we wish to get the value for x and y to find the intersection point on the graph, we need to eliminate either x or y.

                              2x – 2y = 8             (i)

                              x + y = 15              (ii)

To get rid of x multiply the (ii) equation by 2, we’ll get

                             2x +2y = 30

Now change the sign of the equations, we’ll get             

                            +2x – 2y = 8                       

                           + 2x +2y = 30 

                            –       –        –


                             0 x – 4y = – 22

                                       y =   5.5

Now put y = 5.5 in either equation (i) or (ii), we’ll get

                         2x – 2(5.5) = 8

                            2x – 11    = 8

                                      x = 9.5

 The solution of the equation is (9.5, 5.5). The graph is given below-

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  1. Substitution method

The substitution method is the method in which we write the equation in one variable and then plug it into another equation to get the value for x and y.

Example 2:     Solve 2x – 2y = 8        (i)

                                  x + y = 1          (ii)

To find the value for x and y, we need to write the above equation in terms of either x or y,

                                             x = 1 – y

Put x = 1 – y in equation (i),

 2(1 – y) – 2y =8

 2 – 2y – 2y = 8

 2 – 4y = 8

        y = – 1.5

Put y = -1.5 in equation (ii),

 x + y = 1

       x= 2.5

 Therefore the solution for the graph is (2.5, -1.5). The graph is given below-

  1. Augmented matrix

The augmented matrix is a quick method in which we write the equation in form of a matrix. In a matrix, the row represents the constants from the above equation and each column demonstrates the coefficient for a respective single variable.

Example 3: Solve for: 3x – 2y =14

                                     x +3y = 1

First of all, we need to write the equation in an augmented matrix

The above methods can also be used when three or more variables are given. The only change will be in the augmented matrix. We need to use Gauss-Jordan elimination to solve a system of three equations.