More About Word Problems

The word problems we looked at in the previous section were obvious. They practically shouted out, "Look! I'm a word problem and I'm talking about two lines!'' Aside from their outbursts being needlessly loud and distracting us from our cross-stitching, it's also a little presumptuous of them to assume we care. Now we're going to get more subtle, not to mention quieter. These problems are more likely to whisper than shout, which is good news for our ears as well as this monogrammed pillowcase we're working on.

When we solve a system of two linear equations, we're using two pieces of information (the two equations) to find two unknown quantities (usually called x and y).

When a word problem contains two quantities we don't know and gives us two pieces of information about these quantities, we can write a system of equations to describe the word problem. Then we can use that system of equations to solve the problem.

There are three main parts to solving one of these word problems.

  1. Describe the word problem using a system of equations.
     
  2. Solve the system of equations.
     
  3. Check your answers.

The first part tends to be the most troublesome, so we're going to worry about it first. We also like to deliver bad news before delivering good news, so if we ever tell you we've got some good news and some bad news, brace yourself.

  • Translating a Word Problem into a System of Equations


    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 4x cat legs. Since a bird has 2 legs, if the lady owns y cats there are 2y bird legs. This means we can replace this second piece of information with an equation:

    4x + 2y = 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 this 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

    2l + 2w.

    The first piece of information can be represented by the equation

    2l + 2w = 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

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

    or

    4l + 2(w + 3).

    The second piece of information can be represented by the equation

    2l + 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.

  • Solving Word Problems with Systems of Equations


    The best way to learn how to solve a word problem is to do it. Same goes for freestyle swimming, riding a bike, or writing a book about the ideological origins of the American Revolution. Okay, actually the last one could benefit from some preliminary research. You may not want to just dive in on that one.

    We've already done the translation for each of the exercises in this section, so don't start claiming "déjà vu" if they seem familiar. It can't hurt to go through them again and solve them this time, so head on over to the "Exercises" section on the left side of this page. You've got this.