Linear Equations With Integers

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One of the most useful algebra rules on the GMAT is that, to solve for two variables, you need two distinct linear equations. However, there are exceptions to every rule, especially with the real-world constraints of word problems.

To see what I mean, let's look at an example.

7x + 5y = 29

There are an infinite number of solutions to this equation. If x = 1, y = 22/5. If x = 9/7, y = 4, and so on. Positives and negatives, integers and fractions, all are acceptable.

Now consider a very similar problem:

Frida buys x books for $7 each and y books for $5 each. If she spends a total of $29 on books, what is the value of x?

If you translate that to algebra, the result looks familiar:

7x + 5y = 29

But now, the question makes it sound like there's just one value for y. What's different?

When Variables are Integers

When a word problem refers to a number of people, or books, or tickets, or any number of other common units, we need an integer. After all, we can't have 7.36 people or 22/5 books. Thus, when we solve 7x + 5y = 29 for the second time, we're limited to integers for x and y.

The most thorough way to do this is to consider all of the possible values for one of the variables, then see what the other variable must be. Start with the variable with the larger coefficient, in this case y. The largest possible value of y is 4, since if y is 5, 7y = 35, which is larger than 29. If Frida spent $29 on books, she couldn't have spent $35 on just some of the books.

We have five possible values of y, each of which implies a corresponding value of x:

  • y = 0, then x = 29/5
  • y = 1, then x = 22/5
  • y = 2, then x = 3
  • y = 3, then x = 8/5
  • y = 4, then x = 1/5

Only one of those options includes an integer value for both variables, so that must be our answer.

Spotting and Solving

Questions like this one look a lot like questions that have multiple solutions, so you must be careful. The most important characteristic of these problems is what we've already discussed: The variables represent something that can only be denominated in integers.

However, that doesn't always mean there's only one solution. In the equation 2x + y = 8, there's more than one possibility where both variables are integers:

  • x = 0, y = 8
  • x = 1, y = 6
  • x = 2, y = 4, etc.

Another important characteristic, then, is that the coefficients have few common factors. In the first example (5x + 3y), 5 and 3 have no common factors. In the second (2x + y), one coefficient is a multiple of the other. For a good example of large coefficients, see Data Sufficiency #123 in the 12th edition of the Official Guide.

The most devious thing about questions like this is that they look so simple. At first glance, DS #123 looks very simple; it's the sort of question that, if you saw it late in your exam, you might suspect you weren't doing very well. The GMAT doesn't just test your ability to handle ugly, complex problems, it also wants to know how well you can spot exceptions like this one.



About the author: Jeff Sackmann has written many GMAT preparation books, including the popular Total GMAT Math, Total GMAT Verbal, and GMAT 111. He has also created explanations for problems in The Official Guide, as well as 1,800 practice GMAT math questions.

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