Each set of prices can be described by a linear function. In this case, Company A would be y = 5 + .5x, where y is the fare and x is the number of miles driven. Similarly, Company B would b y = 10 + .1x. The definition of intersection tells us that two lines interstect only at that point where their x and y coordinates are identical, which means, of course, that their y coordinates will be identical. So to find the point of intersection we can examine the point where the y coordinates are the same. Which is to say, where 5 + .5x = 10 + .1x. We can simplify this to .5x - .1x = 10 - 5, then .4x = 5, and finally, x = (5/.4) = 12.5. In other words, the fares will be equal when the miles driven is 12.5
