solve a system of equations

Question

Two groups of workers are painting a bridge in the bay. The first group is responsible for painting the north side of the bridge, and the second group is responsible for painting the south side of the bridge. The first group has already painted 4 kilometers of the bridge and is painting 1 additional kilometer per day. The second group has already painted 1 kilometer of the bridge and is painting 4 additional kilometers per day. After a while, the two groups will have painted the same amount of the bridge. How long will that take? How much of the bridge will each group have painted?

Answer 1

If you let x be the number of days from now, then the distance painted by the first group (in kilometers) is 4 + 1x, for the 4km already painted plus another 1km per day.

The second group has painted 1km and is painting another 4 km per day, so they can be represented as 1 + 4x. When both groups have painted the same amount, their work will be equal, so 4 + x = 1 + 4x. To solve, subtract 1 from both sides to get 3 = 3x. Now divide both sides by 3 to get 1 = x. This means they will have painted the same amount after another 1 day. To find how much they will have painted, plug the "1 day" that you found into one of the formulas, and get 4 + 1(1) = 4 + 1 = 5. This means that the group will have painted 5 km. You can verify that this solution is correct by plugging the "1 day" into the other formula, to get 1 +4(1) = 1 + 4 = 5.