For most of this unit we'll be concerned with linear systems of equations, or systems of equations in which all the equations are linear.
These systems are nice because we can graph each equation as a line. A solution to a system of linear equations is a point which lies on all lines in the system. Funny...we just got off the phone with a telemarketer who was also lying on the line.
We will mostly be looking at systems that have only two equations, meaning we're looking at only two lines at a time. We're starting off simple because we don't want you to get dizzy. We love watching people fall down as much as the next website, but we don't want it to happen to you.
There are three things that can happen when we graph two lines.
1. The lines may intersect exactly once.
2.The lines may be parallel (have the same slope) and not intersect at all.
3. The lines may be the same and intersect at every point.
A solution to a system of two linear equations is a point that lies on both lines at once. If the lines intersect exactly once, the system has one solution. If the lines don't intersect at all, the system has no solutions. If the lines are the same line, the system has infinitely many solutions. If the lines decline comment and refuse to tell us how many times they intersect, they are probably hiding something. We should launch a full-scale investigation.
These are the only possibilities for a system of two linear equations: we either have 1, 0, or infinitely many solutions, with the solutions being points. Note that it's a different story if we're able to use curved lines, but here we are dealing only with linear equations. Curved lines have not been invited to this party. They put a hole in the living room wall last time they were over, and they won't be invited back.
To solve a system of equations means to determine if the system has 1, 0, or infinitely many solutions, and to find the solution(s) if they exist. If they don't exist, you can write a short sci-fi story about them.
There are three main ways to solve a system of linear equations:
Be careful: Whichever method you use to solve a system of equations, check your answer to make sure it really is a solution to the system of equations. Some decoys will masquerade as correct solutions only to gain and exploit your trust. Until you have double-checked, don't trust any solution further than you can throw it. Which, considering that we're dealing with abstract mathematical concepts and not tangible objects, isn't far.