Solving Systems of Equations

A system of equations consists of more than one linear equation each containing the same two variables.  The solution of a system is the ordered pair or pairs which make both of the linear equations true at the same time.  To determine if an ordered pair is a solution to a system, the ordered pair must “work” in all equations in the system.  For example, if (3, 4) satisfies both 2x + y = 10 and 3x + 2y = 17, then (3, 4) is said to be the solution to that system of equations.  Since 2(3) + 4 = 10 is true and 3(3) + 2(4) = 17 is true, then (3, 4) is the solution to the system.  Try (4, 3) in the system, x + 2y =10 and 3x + 5y = 3 … (4) + 2(3) = 10 is true, 3(4) + 5(3) = 3 is false; therefore, (4, 3) is not a solution to the system since
(4, 3) does not satisfy both equations.  Solving a system of two equations involves finding the ordered pair or pairs that “work” in both equations.  There are three ways of solving systems of linear equations:
1) graphing, 2) substitution, and 3) elimination (or adding).

There are three possible solutions to a system of equations:
First, if the lines have different slopes then they will have one solution.  The lines when graphed will intersect in one point.  These lines are said to be consistent.
Second, if the graphs are the same … same slope, same y-intercept … then the lines have infinite solutions since every point on one line will coincide with all points on the “other” line.  These lines are said to be dependent.
Third, if the lines have the same slope, but different y-intercepts, then the lines are parallel.  Since they are parallel, they will never intersect and will not have a solution.  These lines are said to be inconsistent.

Graphing a system of equations allows a “picture” of the solution.  If two linear equations are graphed on the same axis then their solution is the point of intersection. You may review graphing lines.

If you graph two lines onto one coordinate axis and note the intersection, if any, then you will have solved that system.  Graphing systems of equations to determine a solution is a very inaccurate method, especially when there are fractional answers.  The remaining two methods Substitution and Elimination are algebraic and very accurate.

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