System of Equations: Types of Solutions

A Graphical Approach

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).

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

(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.

How do we determine parallel lines?

• parallel lines have the SAME slope.

• parallel lines have DIFFERENT y-intercepts.

​No Solutions

How do we determine lines are Identical?

• parallel lines have the SAME slope.

• parallel lines have SAME y-intercepts.

​Infinite Solutions

Linear Systems can have the following types of solutions