Linear equations are the equations of straight lines. The combination of such lines is called linear combination.  It is a method used in solving the given system of linear equations.  


The following are the steps involved in solving system of equations using linear combination method.

  • The equations are arranged with the like terms in columns that is if the equations consists of the variables ‘x’ and ‘y’, the ‘x’ terms, the ‘y’ terms and the constant terms are lined up in the vertical way that is like columns then a line is drawn under them so that the equations can be added
  • Make one set of the terms same with opposite signs so that one variable can be eliminated
  • The like terms are added
  • The equation is solved for the second variable  
  • This value of the variable is substituted in either of the equations and solved for the variable which got eliminated in the second step
  • Finally the x and y values are written as an ordered pair which would be the solution set
  • The solution set is checked if true by substituting the values of x and y in both the equations

Thus using this method we can solve the given system of equations


Let us now solve the given linear equations using Linear combination examples.

Linear Combination Problems:

Solve y-x=-6 and x+y=18 using the linear combination method.

Given equations are, y-x=-6 and x+y=18. The first equation can be re-written as x-y=6

Step1: Arranging the like terms in columns

                x – y = 6

                x + y = 18

Step2: In the above equations the ‘y’ term is the same with opposite sign and hence can be eliminated

             by adding the like terms

                x – y = 6

                x + y = 18

             2x      = 24

Step3: Solving for x by dividing on both sides with 2

                2x/2 = 24/2

                x = 12



Step4: Now that we got the value of ‘x’, we can substitute this value in one of the equations and solve

             for ‘y’

                let us consider the equation x – y = 6

                                                                       12 – y = 6

                Subtracting 12 on both sides

                                                12 – y -12 = 6 – 12

                                                -y = -6

                Multiplying on both sides with -1

                                                -y (-1) = -6 (-1)


Step5: The solution set is (x,y) = (12,6)

Step6: Checking the solution set by substituting the values in both the equations

                LHS = x-y=12-6 =6= RHS

                LHS= x+y=12+6 = 18= RHS, so the solution set is correct!