Systems of Linear Equations

*A system of linear equations is a set of two or more linear equations in the same variables.

*A system of linear equations must be solved at the same time, or simultaneously.

See the example to the right. This is an example of a system of linear equations.

*A solution to a system of linear equations in two variables is an ordered pair that is the solution of each equation.

In the case of our example when x = -1 and y = 2, both equations are true. The solution is (-1, 2).

Is (-1, 2) a solution to this example?

Let's find out:

1. Plug -1 in for x in both equations.
2. Plug 2 in for y in both equations.
3. Ask yourself, "Are both of the equations true?"
4. If the answer is yes, then this is a solution.

In the case of our example, (-1, 2) is a solution.

Another way to find out if a set of points, (x,y) is a solution is by looking at the graph.

Where ever the graphs of linear functions intersect we have a solution.

In this case we have one solution for the system y = -x + 7 and y = 2x + 1 at the point (2, 5).

When lines never intersect, then there are no solutions to the system of equations.

Notice in the example that the lines have the same slope and different y-intercepts. (Recall: y=mx+b)

Notice in the example that the equations look different at first, but after a little algebra, we see that they are the same line. There are infinitely many solutions.