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Lots of word problems can be solved with systems of linear equations. However, before we bother with those, let's look at some word problems that describe single lines, and we're not referring to orderly rows of petrified eighth graders on their way into a school dance.
The hardest part of word problems is usually translating from English into math, so we'll practice that part first.
Write and graph the linear equation described by the following statement:
Jenna works at a retail shop. She makes $10 per hour, plus $3 for each item she sells.
The amount Jenna makes depends on how many items she sells, so our independent variable x should be the number of items she sells, and the dependent variable y should be the amount she's paid. On the graph, the horizontal axis will represent the number of items Jenna sells during one hour, and the vertical axis will represent the amount she gets paid during that hour. By the way, $3 is quite a commission rate, considering that most of the shop's inventory consists of cheap polyester scarves that sell for $10 a pop. Her dad must run the place.
Since Jenna is paid $10 if she sells 0 items, the point (0, 10) will be on the graph. If Jenna sells 1 item she's paid $13, and if she sells 2 items she's paid $16, so the points (1, 13) and (2, 16) are also on the graph:
Now we can connect the dots with a line:
Since it doesn't make sense to have Jenna sell a negative number of items, we only draw the part of the line where x is at least 0. We aren't accounting for the dozens of scarves that Jenna has been sneaking into her bag for herself, in which case she is practically selling a negative number of them, but in the interest of simplicity, let's look the other way and pretend we saw nothing.
From the graph, we can now write an equation for this line in slope-intercept form. The y-intercept is 10, and the slope is 3, so the equation we want is
y = 3x + 10.
If we want to be fussy, we can also write down the condition
x ≥ 0.
Whenever we're writing an equation for a word problem, we need to specify what the variables are.
Write and graph the linear equation described by the following statement:
Lukas left his house at noon to go for a drive. He drove at a constant speed. In fact, it was so constant we're not entirely convinced he didn't just set a brick on the accelerator, then recline his seat and take a nap. Anyway, Lukas was 200 miles from home at 3 p.m.
Lukas's distance from home depends on how long he's been driving. Let's have x be the number of hours Lukas has been driving, and
y be Lukas's distance from home.
The graph set-up will look like this:
Since Lukas left from his house, he was 0 miles from his house after 0 hours of driving, so the point (0, 0) is on the graph. After 3 hours of driving, Lukas was 200 miles from his house, so the point (3, 200) is also on the graph. We wonder if all this algebra-talk is distracting Lukas from how homesick he's feeling right about now.
Now we connect the dots to make a line:
As for the equation, we can see that the slope of the line is
Since the y-intercept is 0, the equation of the line is
Again, notice that we only graphed half of the line. It doesn't make sense to think about Lukas driving for a negative number of hours, so we leave that part out. He may be speedy, but he's not quite speedy enough to turn back time. No matter how much Cher wishes he could.
Now that we've practiced turning words into linear equations, let's actually solve a couple of word problems. This is usually a three-step process:
The fourth step, "take a nap," is totally optional.
Jenna works at a retail shop. Yes, she still works there, even after all her thievery, but she'll tell you it has nothing to do with her old man owning the joint. She still makes $10 per hour, plus $3 for each item she sells.
This word problem is describing a line with an equation we found earlier: y = 3x + 10, which describes the amount of cash (y) she makes in an hour if she sells x items.
Since we've found the linear equation, now we can answer the questions.
1. How much does Jenna make in one hour if she sells 5 items during that hour?
Since x is the number of items Jenna sells during one hour, we've got x = 5.
Then y = 3(5) + 10 = 25, which means Jenna would be paid $25. This amount doesn't include tips. Yeah, she makes tips, too. What can we say, this girl knows how to turn a buck.
2. How many items would Jenna need to sell in an hour to make $43 during that hour?
Since y is the amount Jenna is paid, if Jenna makes $43, we want to have y = 43. Plug that into our equation:
43 = 3x + 10
We can solve this equation for x to find:
Since x is the number of items Jenna sells during an hour, in order to make $43 Jenna must sell 11 items. Given her foolproof sales technique of breaking down into tears whenever someone decides not to buy something, she shouldn't have any problem hitting that mark.
Let's check that this is correct, though: If Jenna sells 11 items she'll make 3(11) + 10 dollars, which is indeed $43.
Marcio spent $7 per day. Knowing Marcio, he probably spent it on Lotto scratchers. After five days, he had $8 left. How much money did Marcio start with?
First, we need to come up with a linear equation. The amount of money Marcio has depends on how many days have passed. Let's have x be the number of days that have passed, and y be the amount of money Marcio has.
The statement "after five days, he had $8 left'' tells us that the point (5, 8) is on the graph. It also tells us he "shockingly" hasn't struck it rich yet, or he probably would have given up on these silly things by now.
Since Marcio is spending $7 per day, the slope of the line is -7. We can use this information to find an equation for the line. Let's use point-slope form, since we have a point and a slope. We find the equation:
y – 8 = -7(x – 5)
Now we can worry about answering the question. The amount of money Marcio started with is the amount of money he had when 0 days had passed. Oh, to go back in time and have all that hard-earned cashola back, eh, Marcio?
We want to find the y-intercept of the line, since that's the point where x = 0 days. We can do this by rearranging our point-slope equation into slope-intercept form.
The y-intercept is 43, which means Marcio started with $43. Hey...that's how much Jenna made from selling her 11 items. These two might be in cahoots...
Let's make sure we're right. If Marcio started with $43 and spent $7 per day, after 5 days he would have 43 – 5(7) = 43 – 35, which is indeed 8 dollars.
Word problems that involve a linear equation can give us the information we need to write that equation in several different ways. We could be told two points on the line, or a point and a slope, or the y-intercept and the slope, or both intercepts. We could be given a treasure map that will lead us to the information we need, although those problems are more rare. Word problems can ask questions about the intercepts of the line or the slope. They can provide one coordinate of a point on the line and then ask for the other coordinate.
Come to think of it, they ask us for a whole lot of stuff without giving much back in return. We're in a one-sided relationship and should probably get out of it. We'll see what our therapist has to say about this on Tuesday.
After we find the line described by the word problem, the trick, as usual, is to figure out what the question is actually asking. Don't be distracted by any of its mumbo-jumbo.
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:
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.
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.
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.