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 $t = 0$, then the distance in miles it has traveled from the rest area after $t$ hours is

$\displaystyle d = 75t$

The point $(2, 150)$ is on the graph of this equation because it makes the equation true ($150 = 75 \boldcdot 2$). 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

$\displaystyle d = 14 - 65t$

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 $t=0$ and travels in the same direction as Car A, also going 75 miles per hour. It's equation is also $d=75t$. 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 $y = 3x+2$ and $2y = 6x +4$, 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.