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A Graphical Approach

(Infinitely Many Solutions)

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.

A Graphical Approach

(No Solutions)

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)

A Graphical Approach

(One Solution)

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).

Fill in the Blanks

Solutions 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).

What is a system 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.