Solving Systems of Equations by Elimination

To solve a linear system of equations by the method of elimination:

1)  write both of the equations in standard form, x term and y term on the left-hand side of the equation and constant term on the right-hand side of the equation,
2)  place the equations in column form (one equation under the other) and “pre-add” (in your head) the x-terms and the y-terms from both equation to see if one or both of the “like” terms will be eliminated …
if neither the x terms, nor the y terms will be eliminated, then you will have to multiply one or both of the equations by some integer so that the coefficients of either the x terms or the y terms will be additive inverses … once you have made appropriate changes to one or more of the lines by multiplying then actually add the two equations, eliminating either the x terms or the y terms,
3)  step 2 will result in an equation in one variable, solve this equation for the variable,
4)  use the value found in step 3 to find the value of the remaining variable, you may use either equation to find the second value,
5)  check the solution in both equations to be sure that the solution works in both equations.

Solve the system of equations … 3x – 2y  = 19 and x + y = 8 … by elimination.
(Notice that this is the same system that I solved previously by substitution.  I want to show you the differences in the two methods.  Actually, only the first two steps are different.  Steps 3-5 are the same for both methods.)

     3x – 2y  = 19
        x + y    = 8  … “pre-adding” I would get 4x – y = 27, so I must multiply one of the lines by some integer so that the coefficients of one of the variable terms are additive inverses.  I choose to eliminate the y terms since multiplying the second line by 2 would make – 2y and + 2y additive inverses.

     (2)x +(2)y    = (2)8 … 2x + 2y = 16 … now place the equations once again in column form and add

     3x – 2y = 19
     2x + 2y = 16
     5x         =  35

     x  =  7  … now use this value and substitute it into either of the original equation to find the value of y.

     3(7) – 2y = 19

      21 – 2y  = 19

           – 2y  = 19 – 21

           – 2y  =  –2

             y = 1 … the solution set is the point ( 7, 1 )

Check:  the solution set should work in both of the equations 3x – 2y  = 19 and x + y = 8.
Since 3(7) – 2(1)  = 19 and (7) + (1) = 8, then the point ( 7, 1) is the solution to this system of equations.

General Algebra Tips

The views and opinions expressed in this page are strictly those of Mary Lou Baker.
The contents of this page have not been reviewed or approved by Columbia State Community College.

This page was edited on 19-Sep-2007