## Nature of the Solutions of Systems of Linear Equations

When we have two lines that are graphed on the same plane, one of the following three things can happen:

1. The two lines are actually the same line.
2. The two lines run parallel to each other and never cross.
3. The two lines intersect at one point.

For example, consider the following graph:

There are three lines graphed in this image:

Red: y = -3x + 8
Blue: y = (3/2)x + 1
Green: y = -3x + 4

The red and green lines have the same slope, and so they run parallel to each other. The blue line has a different slope than the red line, so they intersect at some point. The blue line has a different slope than the green line as well, so they also intersect at one point.

If two lines have the same slope and the same y-intercept, then they are the same line and so they intersect at infinite points.

If two lines have the same slope and a different y-intercept, then they are parallel lines and never intersect.

If two lines have different slopes, then they intersect at one point, no matter what the y-intercepts are.

All of this depends on us having the equations in slope-intercept form. So before you can solve a system of linear equations by graphing, you have to get them into slope-intercept form.

One possible thought process once you get the lines in slope-intercept form is the following:

1. Do the lines have the same slope?

- If not, then they intersect at one point, and we’ll be able to find that point.
- If so, then we proceed to step 2.

2. Do the lines have the same y-intercept?

- If not, then they are parallel lines, and there are no solutions because they never intersect.
- If so, then they are the same line, and there are infinite solutions because they intersect at every point on the line.

If you have a set of equations that intersect at one point, then you can solve for that point by graphing, using substitution or using elimination.