# Systems of Linear Equations

### Topics

## Introduction to :

To describe a word problem using a system of equations, we need to figure out what the two unknown quantities are and give them names, usually *x* and *y*. We could name them Moonshadow and Talulabelle, but that's just cruel. Next, we need to use the information we're given about those quantities to write two equations.

### Sample Problem

Set up a system of equations describing the following problem:

A woman owns 21 pets. Each of her pets is either a cat or a bird. If the pets have a total of 76 legs, and assuming that none of the bird's legs are protruding from any of the cats' jaws, how many cats and how many birds does the woman own?

There are two unknown quantities here: the number of cats the lady owns, and the number of birds the lady owns.

The problem has given us two pieces of information: if we add the number of cats the lady owns and the number of birds the lady owns, we have 21, and if we add the number of cat legs and the number of bird legs, we have 76.

Let's replace the unknown quantities with variables. Let

*x *be the number of cats the lady owns, and *y *be the number of birds the lady owns.

Now we can replace the pieces of information with equations. Instead of saying "if we add the number of cats the lady owns and the number of birds the lady owns we get 21, " we can say

*x* + *y* = 21.

What about the second piece of information: "if we add the number of cat legs and the number of bird legs, we get 76?" Since a cat has 4 legs, if the lady owns *x* cats there are 4*x* cat legs. Since a bird has 2 legs, if the lady owns *y* cats there are 2*y* bird legs. This means we can replace this second piece of information with the equation

4*x* + 2*y* = 76.

If *x* is the number of cats and *y* is the number of birds, the word problem is described by this system of equations:

In the previous problem, *x* meant the number of cats and *y* meant the number of birds. In order to have a meaningful system of equations, we need to know what each variable represents. If we can master this skill, we'll be sitting in the catbird seat.

Let's do some other examples, since repetition is the best way to become fluent at translating between English and math. Well, that or spending a semester studying abroad in Mathrovia.

The problems are going to get a little more complicated, but don't panic. It just means we'll see more variety in our systems of equations.

### Sample Problem

Write a system of equations describing the following word problem:

The Lopez family had a rectangular garden with a 20 foot perimeter. They enlarged their garden to be twice as long and three feet wider than it was originally. They had to, since their cherry tomato plants were getting out of control. The enlarged garden has a 40 foot perimeter. What were the dimensions of the original garden?

The problem asks "What were the dimensions of the original garden?" meaning that the two unknowns we're looking for are the length (*l*) and width (*w*) of the original garden:

Our first piece of information is that the original garden had a 20 foot perimeter. From looking at the picture, we can see that the perimeter is

*l* + *l* + *w* + *w*

or

2*l* + 2*w*.

The first piece of information can be represented by the equation

2*l* + 2*w* = 20.

Our second piece of information is that if we make the garden twice as long and add 3 feet to the width, the perimeter will be 40 feet. The new garden looks like this:

The perimeter of this new garden is

2*l* + 2*l* + (*w* + 3) + (*w* + 3)

or

4*l* + 2(*w* + 3).

The second piece of information can be represented by the equation

2*l* + 2(*w* + 3) = 40.

To sum up, if *l* and *w* are the length and width, respectively, of the original garden, then the problem is described by the system of equations

Some day, you may be ready to determine the length and width of an Olive Garden. We'd be dealing with some large numbers, though. You need a lot of room if you're going to be storing endless breadsticks.

#### Example 1

Set up a system of equations describing the following word problem: Whenever Anna drives, she goes one speed. Whenever Bob drives, he drives one speed. Anna and Bob went on a road trip together. They insist it is strictly platonic, but that goo-goo-ga-ga look in their eyes says differently. The first day Anna drove for 2 hours, Bob drove for 3 hours, and they went 340 miles. The second day Anna drove 4 hours, Bob drove 1 hour, and they went 330 miles. How fast does Anna drive? How fast does Bob drive? Will they or won't they? |

#### Exercise 1

Set up a system of equations describing the following word problem. Be sure to specify what each variable represents.

A forest inhabited only by borogoves and mome raths contains eighty-two creatures. Borogoves are two-legged animals, while a mome rath has 4 legs. Sorry, we were getting bored of cats and birds.

If the forest contains a total of 234 legs, how many borogoves and how many mome raths live in the forest?

#### Exercise 2

Set up a system of equations describing the following word problem. Be sure to specify what each variable represents.

Stanley has $1.45 in dimes and quarters. If he has 10 coins total, how many of each kind of coin does he have?

#### Exercise 3

Set up a system of equations describing the following word problem. Be sure to specify what each variable represents.

A private school spent $860,000 in one year to pay all of its teachers and administrators. The school has a total of twenty-four teachers and administrators combined. If teachers make $35,000 per year and administrators make $40,000 per year, how many teachers and how many administrators does the school have? Also, where in the country do these people live that they are able to get by on such a pittance? You hear that, private school? We're calling you out.

#### Exercise 4

Tom and Jerry drove 270 total miles in 5 hours. Tom always drives 40 mph and Jerry always drives 60 mph. It's a wonder they can manage to remain friends. How many hours did Tom drive and how many hours did Jerry drive?

#### Exercise 5

Blue beads cost $0.50 per ounce and green beads cost $0.75 per ounce. Janine wants to mix blue and green beads to get 10 ounces of a bead mixture worth $0.55 per ounce. Wow...Janine has some extremely specific designs on these beads, it seems. Hey, good for her for knowing exactly what she wants.

How many ounces of each color does Janine need to buy?

#### Exercise 6

For each problem above, you found a system of equations. What did these systems of equations have in common?

#### Exercise 7

Ayako spent $7.80 on equal weights of white and red beads. The red beads cost twice as much per ounce as the white beads. Well, sure they do. The world's supply of the color red has recently dipped to an all-time low, and is in high demand.

How much did Ayako spend on each color of beads?

#### Exercise 8

A rectangle has perimeter 18. If each side of the rectangle is doubled, the perimeter of the new rectangle is 36. What were the dimensions of the original rectangle?

#### Exercise 9

5 ounces of dark chocolate and 11 ounces of milk chocolate combine to form one pound of 52.5% chocolate. Eight ounces each of dark and milk chocolate combine to form one pound of 60% chocolate. What percent chocolate are the dark and milk chocolate being used? Bonus question: dark chocolate kicks milk chocolate's behind. Okay, so the bonus question wasn't actually a question, but now you know where we stand.

#### Exercise 10

Jemima wants to make chocolate-chip walnut brownies. Chocolate chips come in a 12 oz bag that costs $3. Walnuts come in a 4 oz bag that costs $2. If Jemima needs three pounds of chocolate chips and walnuts combined, and has $15 to spend, how many bags of each can she buy? BTW, what's with the shoestring budget, Jemima? If you're going to start this chocolate-chip walnut brownie venture of yours, build up some capital and really go for it. Even if you're only entertaining a few guests.