# Systems of Linear Equations

### Topics

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.