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Manipulating Algebraic Equations and Expressions with Fractions: Systems of Equations

A student writes,

Sometimes the answer explanation suggests that I clear fractions by multiplying an entire equation by a common denominator. Sometimes the answer explanation suggests that I manipulate just one side of an equation. How do I know which is better?

This is a surprisingly subtle question. It turns out that the answer is partly “Do what’s mathematically sound,” but mostly “Do what’s useful.” It’ll take me 6 posts to show what that means:

So far, all the questions we’ve considered in this series of posts have been in the Problem-Solving format. Things can get a little more complicated in the Data Sufficiency format, because questions in that format often present us with systems of equations.

Consider this question:


It’s tempting to suppose that the correct answer is D, since the “if” statement provides one equation and each of the numbered statements provides an equation. However, it’s not the case that just any pair of equations yields values for two variables. The rule is that to solve for two (or n) variables requires two (or n) distinct linear equations. In this case, the equation xy=10 is not linear.

Let’s suppose that you decided to start with Statement (2). Before solving algebraically for the values of the variables (which approach would require first constructing and then solving a quadratic equation), check whether simple integer values fit both Statement (2) and the “if” equation. After all, the GMAT usually uses pretty manageable values, and there are too many possibilities for the two relevant simple equations.

In fact, it looks as though x and y could be 2 and 5. But which is which? Since y -x could be either 5-2 or 2-5, Statement (2) is not sufficient. Eliminate answers B and D.

Statement (1) is more interesting still. In the last post in this series I wrote that when faced with an algebraic equation involving fraction, you should generally multiply through by the least common denominator to clear all the fractions. Questions which ask you for the value of an expression with more than one variable are sometimes exceptions. In this case, for instance, it’s likely easier to solve for y – x

directly than for y and x separately.

This means that rather than multiply through by 10xy to clear the fractions, we should manipulate the equation to isolate y – x.


Cross multiply the terms on the left-hand side of the equation.


Multiply each side of the equation by xy to clear the fraction on the left-hand side.


It looks as though our question, “What is  y – x?”is equivalent to the question, “What is xy?” Since Statement (1) provides and answer to that question, Statement (1) is sufficient. The correct answer is (A).

It turns out that many Data Sufficiency problems with systems of equations work just as this one did.


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2 Responses to Manipulating Algebraic Equations and Expressions with Fractions: Systems of Equations

  1. Guest October 2, 2015 at 7:27 pm #

    Hi, but how statement 1) can be sufficient if we don’t know xy?

    • Thomas Lawson March 21, 2016 at 7:50 pm #

      It gives us xy=10 in the prompt of the question, so he manipulated the equation to properly get xy into the problem. He failed to elaborate a little more to make it absolutely clear, because in data sufficiency you do not need to solve the problem, but rather know that you can. To solve the problem however, you insert 10 to replace xy which when multiplied by 3/10 leaves you with y-x=3 which is the original question, what is the value of “y-x?”

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