Now we'll move on to some word problems that involve finding where two lines intersect—in other words, solving a system of equations. We'll start with word problems that are obvious about describing two distinct lines. You have been warned. The following word problems are not notorious for their subtlety.
The overall strategy is the same here as it was when the problems had only one line:
- Find equations for the lines.
- Figure out what the question is asking so we can answer it.
- Check the answer.
Sample Problem
Tammy and Lisa work in retail, in different shops. Tammy makes $7 per hour plus $4 for each item she sells, and Lisa makes $10 per hour plus $3 for each item she sells. Their friend Julie makes $12 per hour plus $2 for each item she sells, but Julie isn't even a part of this word problem, so don't worry about that. In fact, we're not even sure why we bombarded you with that extraneous information. We just thought you might like to know what Julie's up to.
How many items would Tammy and Lisa need to sell in one hour in order for both girls to make the same amount of money during that hour?
This problem is describing two lines. For each line, the independent variable (x) is the number of items sold, and the dependent variable (y) is the amount of money the seller makes.
Let's do the problem by graphing first, to get a sense of what's going on. Tammy starts out making $7:
If she sells 1 item, she makes 7 + 4 = 11 dollars:
If she sells 2 items she makes 7 + 4 + 4 = 15 dollars, and so on:
Lisa, on the other hand, starts out at $10 per hour:
For each item Lisa sells, she gets another 3 dollars, so we get another line:
From looking at the graph, we can see that Tammy and Lisa will make the same amount of money when they each sell 3 items. It'll probably never happen, though, since Tammy is a much better saleswoman than Lisa. You should hear her in action: "What will it take to put you in this foot spa today?" What a pro.
Now let's do the same problem in symbols. We need to find the two equations described by the problem.
Tammy's line has a y-intercept of 7 and a slope of 4, so her equation is
y = 4x + 7.
Lisa's line has a y-intercept of 10 and a slope of 3, so her equation is
y = 3x + 10.
We've found a system of equations:
The problem asks how many items sold will give Tammy and Lisa the same amount of money, so we want to find x when the y-values on the two lines are the same. That is, we want to find where the lines intersect, which we can do by solving the system of equations. Why did one line cross the other line? To get to the other side. Oh...you've probably heard the version with the chicken.
For this system, it's easiest to use substitution. Since y = 4x + 7, we have
We think if each girl sells 3 items, they'll be paid the same. Let's check this: if Tammy sells 3 items, she makes 7 + 3(4) = 19 dollars.
And if Lisa sells 3 items she makes 10 + 3(3) = 19 dollars. It's a match.
Sample Problem
Loren and Marisol each bought cookies on Sunday and started eating their cookies the next day. They're trying to eat as many of them as possible before they need to give cookies up entirely for Lent.
Loren took 6 days to eat 30 cookies, while Marisol took 8 days to eat 24 cookies. On what day of the week did they have the same number of cookies still left to be eaten? Hint: it was the day they both had to be rushed to the local ER to have their stomachs pumped.
The number of cookies depends on the number of days, so let's have x be the number of days that have passed since Sunday, and y be the number of cookies left at the end of that day.
Loren started with 30 cookies. He took 6 days to eat those cookies, which means he must have eaten 5 cookies per day. So much for his diet; that New Year's resolution didn't last very long.
Loren's cookie equation is y = 30 – 5x.
Marisol started with 24 cookies and took 8 days to eat them, so she must have eaten 3 per day. She's not scarfing them at the rate that Loren is, but she could still stand to slow down the assembly line into her mouth a bit.
Marisol's cookie equation is y = 24 – 3x.
We want to know on what day they have the same number of cookies, so we need to find the value of x for which the lines intersect. Again, we can use substitution, in the same way that Loren and Marisol should have considered substituting celery sticks for cookies. Since we have two different expressions for y, we set those equal to each other:
We're almost done, but let's read the question again. We want to know "on what day of the week" Loren and Marisol have the same number of cookies left. We're not sure why we want to know what day of the week...are they going to achieve any more of a sugar high on a Friday than they will on a Tuesday? Anyway, we're not in the business of asking questions. We're in the business of answering them.
We've found a number, 3, and we need to translate that number into a day of the week. Since x is the number of days that have passed since Sunday, x = 1 would be Monday, x = 2 Tuesday, and x = 3 Wednesday. The answer seems to be Wednesday, but let's check to be sure.
Loren eats 5 cookies per day, so he's eaten 5 by the end of Monday, 10 by the end of Tuesday, and 15 by the end of Wednesday, leaving him with 30 – 15 = 15 cookies.
Marisol eats 3 cookies per day, so she's eaten 9 by the end of Wednesday, leaving 24 – 9 = 15 cookies. Now we can be confident that Wednesday is the right answer. It's nice to be confident; confidence is attractive.
The moral of the last problem is to be sure you answer the question that's being asked. We think Aesop had some fable about it, but we're not sure.
Sometimes this sort of problem doesn't have an answer: if the lines are parallel, or if they intersect at a point that doesn't make sense for the problem, then the problem doesn't get a numerical answer. It also doesn't "get" most David Lynch films, but we can't really blame it.
Sample Problem
Lois and Joseph start saving pennies on New Year's Day. Lois has 5 pennies and saves 6 pennies every day. Joseph starts with 4 pennies and saves 2 pennies every day. This situation is actually kind of embarrassing, considering that both are in their thirties. Such is the life of an actor...
After how many days will they have the same number of pennies?
Since Lois starts with 5 pennies and saves 6 per day, her line is y = 5 + 6x.
Since Joseph starts with 4 pennies and saves 2 pennies per day, his line is y = 4 + 2x.
Let's solve the system of equations:
Using substitution, we get:
However, there's something funny going on here: the lines intersect at 
, which would be a quarter of a day in the past. Since Lois and Joseph weren't saving pennies a quarter of a day in the past, the answer 
doesn't make sense in this problem. Lois and Joseph's lines will not intersect for any values of x that aren't negative, so Lois and Joseph will never have the same number of pennies.
They'll also never have the same number of roaches infesting their respective hovels. Don't despair completely for them; Joseph has an audition for a student film next week, and Lois has a friend who's agreed to give her free headshots. Next stop: superstardom.
We could also answer this question without doing any arithmetic. Lois starts out with more pennies than Joseph does, and Lois also saves more pennies per day than Joseph does. There's no way Joseph could ever catch up to get as many pennies as Lois has. Not that Lois is much better off than Joseph. After all, neither of them can afford food.