This section explains the different methods used to solve systems of linear equations algebraically.

There are essentially three different methods to solve systems of equations algebraically. They are listed and described briefly below.

1. The Graphing Method: There is one variable solved in both equations; therefore, it is easy to use a calculator for problems like these:
   

   m = -3n
   m = 10 – 4n

   Substitution is used to check the problem.
2. The Substitution Method: There are two different types of equations where substitution would be the easiest method.
   1. One variable is solved for in one of the equations. The system is solved by substituting the equation with the isolated term into the other equation:
   

   x + 2y = 7
           y = x – 5

   Graphing is used to check the problem.
   2. One variable can be easily solved. The system is solved by solving for one variable in one of the equations then substituting the equation into the second equation: (Solve for a in the second equation then substitute the second equation into the first equation.)
   

   2a + 3b = 2
    a – 2b = 8

   Graphing or elimination is used to check the problem.
3. The Elimination Method: Both equations are in standard form, Ax + By = C. The system of equations are solved by eliminating a variable then solving for the remaining variable (Add the two equations together to eliminate the y then solve for x.) :
   

   8x + 11y = 37
   2x – 11y = -7

   Graphing is used to check the problem.