There are two methods that will be used in this lesson to solve a system of linear equations algebraically. They are 1) substitution, and 2) elimination. They are both aimed at eliminating one variable so that normal algebraic means can be used to solve for the other variable. Once one variable is solved, then substitution will be used in both above methods to find the second variable.
Special Circumstances.
When solving inconsistent systems algebraically, you will find that
all variables drop and the remaining statement is false. When you receive
a false statement that has no variables in it, first check your algebra.
If you have done it correctly, and there is a false statement remaining,
then you know that you have no solution, and the system is inconsistent.
When solving consistent dependent systems algebraically, you will find that all variables drop and the remaining statement is true. When you receive a true statement that has no variables in it, first check your algebra. If you have done it correctly, and there is a true statement remaining, then you know that there are an infinite number of solutions, and the system is consistent dependent.
The Substitution Method.
To use the substitution method, you solve one of the equations for
either variable, and then substitute that algebra expression in for the
same variable in the other equation. This will allow you to solve for one
variable. Then you would substitute that value into either original equation
to find the value of the other variable.
Example.










Check this by plotting both lines and visually checking the solution
point.
It does check. 
Example Problem.












Check this by plotting both lines and visually checking the solution
point.
It does check. 