The solutions to an equation correspond to points on its graph. For example, if Car A is traveling 75 miles per hour and passes a rest area when , then the distance in miles it has traveled from the rest area after hours is

The point is on the graph of this equation because it makes the equation true ( ). This means that 2 hours after passing the rest area, the car has traveled 150 miles.

If you have 2 equations, you can ask whether there is an ordered pair that is a solution to both equations simultaneously. For example, if Car B is traveling toward the rest area, and its distance from the rest area is

We can ask if there is ever a time when the distance of Car A from the rest area is the same as the distance of Car B from the rest area. If the answer is yes, then the solution will correspond to a point that is on both lines.

Graph of 2 lines, origin O, with grid. Horizontal axis, time in hours, scale 0 to point 22, by point 0 2’s. Vertical axis, distance in miles, scale 0 to 14, by 2’s. One line passes through the origin and the point 0 point 1 comma 7 point 5. Another line crosses the y axis at 14 and passes through the point 0 point 1 comma 7 point 5  

Looking at the coordinates of the intersection point, we see that Car A and Car B will both be 7.5 miles from the rest area after 0.1 hours (which is 6 minutes).

Now suppose another car, Car C, also passes the rest stop at time and travels in the same direction as Car A, also going 75 miles per hour. It's equation is also . Any solution to the equation for Car A is also a solution for Car C, and any solution to the equation for Car C is also a solution for Car A. The line for Car C is on top of the line for Car A. In this case, every point on the graphed line is a solution to both equations, so there are infinitely many solutions to the question, “When are Car A and Car C the same distance from the rest stop?” This means that Car A and Car C are side by side for their whole journey.

When we have two linear equations that are equivalent to each other, like and , we get 2 lines that are right on top of each other. Any solution to one equation is also a solution to the other, so these 2 lines intersect at infinitely many points.

In general, if we have 2 lines in the coordinate plane and we have their corresponding equations,

• The coordinates of a point on a line make that equation true.
• The coordinates of a point off of a line make that equation false.
• The coordinates of a point that is the intersection of the 2 lines make both equations true.