Solving Systems of Linear Equations Using Elimination

Systems of Linear equations:

A system of linear equations is just a set of two or more linear equations.

In two variables (x and y), the graph of a system of two equations is a pair of lines in the plane.

There are three possibilities:

  • The lines intersect at zero points. (The lines are parallel.)
  • The lines intersect at exactly one point. (Most cases.)
  • The lines intersect at infinitely many points. (The two equations represent the same line.)

How to Solve a System of Linear Equations Using The Elimination Method (aka The Addition Method, aka The Linear Combination Method)

  • Step 1: Add (or subtract) a multiple of one equation to (or from) the other equation, in such a way that either the x-terms or the y-terms cancel out.
  • Step 2: Then solve for x (or y, whichever's left) and substitute back to get the other coordinate.

Example:

Solve the system

Multiply the first equation by –2 and add the result to the second equation.

         –8x – 6y = 4

           8x – 2y = 12

                 –8y = 16

Solve for y.

                     y = –2

Substitute for y in either of the original equations and solve for x.

         4x + 3(–2) = –2

               4x – 6 = –2

                     4x = 4

                       x = 1                                           

The solution is (1, –2).