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Systems of Linear Equations

Systems of Linear Equations


Introduction to :

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:

  1. Find equations for the lines.
  2. Figure out what the question is asking so we can answer it.
  3. Check the answer.

Sample Problem

Tammy and Lia work in retail, in different shops. Tammy makes $7 per hour plus $4 for each item she sells, and Lia makes $10 per hour plus $3 for each item she sells. 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 need to 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 each girl need to sell in one hour in order for Tammy and Lia 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:

Lia, on the other hand, starts out at $10 per hour:

for each item Lia sells, she gets another 3 dollars, so we get another line:

From looking at the graph, we can see that when Tammy and Lia will make the same amount of money when they each sell 3 items. It will probably never happen, though, since Tammy is a much better saleswoman than Lia. 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.

Lia'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 Lia 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,

and if Lia sells 3 items she makes

10 + 3(3) = 19.

Sample Problem

Loren and Marisol each bought cookies on Sunday and started eating their cookies the next day. They are 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 has 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?

This problem is describing two lines where

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

We must have

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 will 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.

Example 1

A brother and sister each grab a few handfuls of candy. The brother takes 17 hours to eat 51 pieces of candy, while the sister takes 9 hours to eat 27 pieces of candy. No, it's not Halloween. Just a normal day in the Howell household. After how many hours did the two siblings have the same amount of candy left?

Exercise 1

Bobby didn't have any cookies, so he started buying two cookies per day. He didn't eat them, only collected them in a box. He used to collect stamps. We don't know why he decided to make the switch to something perishable.

Jen started out with 100 cookies and ate 3 every day. Apparently, she is a collector of calories. Delicious, delicious calories.

After how many days did Bobby and Jen have the same number of cookies?

Exercise 2

Uncle Scrooge has 1 million dollars. If Huey starts with $1,000 and makes $100 per day, how many years will it be until Huey has as much as Uncle Scrooge?

Exercise 3

Nick and Kevin went to climb a mountain, starting from sea level. They didn't care for each other, but their girlfriends insisted they spend some time together so they could bond. However, they never promised they would go up the mountain at the same time. Sneaky. Nick started climbing at noon and was 500 feet up by 1 p.m. Kevin started climbing at 12:15 p.m., and at 12:30 p.m. was 75 feet up the mountain. When did Kevin pass Nick?

Exercise 4

Jin and Li have offices on the same highway. Not on the actual highway. All their papers would blow away. They both work just off the same roadway, however.

At noon they leave their offices and drive towards a local restaurant where they have a lunch meeting. Jin's office is 40 miles east of the restaurant, while Li's office is only 20 miles east of the restaurant. What, there wasn't a Subway close by?

If Li gets to the restaurant at 12:20 p.m. and Jin gets there at 12:40 p.m., at what time do Li and Jin pass on the road?

Exercise 5

Aisha and Jojo each drove South on Route 1, heading for mile 0 in Key West, Florida. At 1 p.m., Aisha was 120 miles away from mile 0, driving 40 miles per hour all the way. Jojo was 112 miles away, and drove 28 miles per hour for the rest of the trip.

  1. At what time did Aisha pass Jojo?
  2. When did Aisha get to mile 0? When did Jojo get to mile 0?

Exercise 6

Sandy sells seashells by the seashore. Lucy also sells seashells by the seashore, but her tongue-twister isn't so twisty. After one week of seashell selling, Lucy has 32 seashells left of her original 74, and Sandy has 34 left of her original 90. After how many days of seashell selling do Lucy and Sandy have the same number of seashells left?

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