Zane and his good buddy Tom are both mechanics at a shop that does oil changes. They are in a friendly competition to see who can complete the most oil changes in one day. Zane has already finished 4 oil changes today, and can complete more at a rate of 1 oil change per hour. Tom just came on shift, and can finish 3 oil changes every hour. Sometime during the day, the friends will be tied, with the same number of oil changes completed. How many oil changes will Zane and Tom each have done? How long will that take? Both men will have completed oil changes in hours.

We need to figure out the linear equation for each man's oil changes per hour, figuring in how many they each have done at the start of this competition. Zane has already done 4 by the time Tom gets started. y is the number of oil changes and x is the number of hours per oil change. If Zane already has 4 done and can get 1 done per hour, his equation is

y = x + 4

Tom hasn't done any yet, but he can do 3 per hour, so his equation is

y = 3x

Where these 2 lines intersect graphically is the solution to this system. The solution is an (x, y) coordinate where x is the number of hour it takes to get y oil changes done. We can do this analytically as well as graphically by solving by substitution.

If Tom's equation is y = 3x, we will take that 3x and sub it into Zane's equation for y:

3x = x + 4

Now solve for x, the number of hours where they will both have the same number of oil changes done.

2x = 4 so

x = 2

This tells us that after 2 hours they will have the same number of changes done. Now use that x and solve for y in either equation.

y = 2 + 4 so

y = 6

The solution as a coordinate pair is (2, 6) which in words is

They both get 6 oil changes done when 2 hours has gone by.

Likewise, we could say it will take 2 hours for each of them to have the same number of oil changes (6).