{"id":1814,"date":"2012-05-31T09:00:19","date_gmt":"2012-05-31T16:00:19","guid":{"rendered":"https:\/\/magoosh.com\/gmat\/?p=1814"},"modified":"2020-01-15T10:50:42","modified_gmt":"2020-01-15T18:50:42","slug":"gmat-quant-how-to-solve-two-equations-with-two-variables","status":"publish","type":"post","link":"https:\/\/magoosh.com\/gmat\/gmat-quant-how-to-solve-two-equations-with-two-variables\/","title":{"rendered":"GMAT Quant: How to Solve Two Equations with Two Variables"},"content":{"rendered":"

Fact<\/strong>: If you have two variables and only one equation, in general you will not be able to solve for the individual values of the variables.\u00a0 You need two separate equations in order to solve for the individual values of each of two variables. (This fact is crucially important for GMAT algebra<\/a> questions on the Quantitative Section.)<\/p>\n

More Advanced Fact<\/strong>: As a general rule, if you wish to solve for the values of n different variables, you need at least n different equations.\u00a0 (If n = 3, this could possibly come into play on 700+ questions on GMAT Quantitative.)<\/p>\n

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A Boon for Data Sufficiency!<\/h2>\n

Just this one mathematical fact, the first one, has powerful implications for hundreds of possible DS questions.\u00a0 Consider this template DS question:<\/p>\n

 <\/p>\n

1) In blah blah blah scenario, x is blah blah blah and y is blah blah blah.<\/p>\n

Statement #1: 2x + y = 6<\/p>\n

Statement #2: x \u2013 y = 6<\/p>\n

 <\/p>\n

Statement #1: two variables, one equation, not sufficient by itself.\u00a0 Statement #2: two variables, one equation, not sufficient by itself.\u00a0 Combined: two equations, two variables, can be solved, sufficient.\u00a0 Done.\u00a0 Answer = C<\/strong>.<\/p>\n

Notice, we had to do a minimum of math to solve this question.\u00a0 Just this one fact can make short work of any of a number of seeming complicated DS questions. \u00a0Often, notice that the “equations” are given in verbal form: one first has to translate, but once you realize the verbal information constitutes an equation, you don’t even need to find the question: you can just use this logic to power through to an answer.<\/p>\n

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Strategy for Problem Solving<\/h2>\n

For DS, all you have to determine is whether you can find an answer.\u00a0 On Problem Solving, and on the occasional Two-Part Analysis in the Integrated Reasoning section, actually have to find the answer.<\/p>\n

Your Algebra Two teacher probably taught you two different ways to solve these equations.\u00a0 For simplicity, I am just going to review one method, the one that is most useful in solving the systems of equations you will see on the GMAT.\u00a0 If you remember the other method, and prefer that, by all means use it.<\/p>\n

The method I am going to review is sometimes called “elimination” or “linear combination.”\u00a0 Here is the strategy.<\/p>\n

Step #1 — Multiply one or both equations so that the coefficients of the same variable are opposites of each other (e.g. +7 and \u20137).<\/p>\n

Step #2 — Add the two equations<\/p>\n

Sometimes you get very lucky because in the equation, as given, coefficients of one variable already are opposites, so you can bypass Step #1 and proceed immediately to Step #2.<\/p>\n

I’ll demonstrate with the two equations in my hypothetical question above.\u00a0 Suppose, in a PS question we were given those two equations (2x + y = 6 and x \u2013 y = 6) and had to solve for either x or y.\u00a0 Here, we are very lucky —- the coefficients of y are opposites, all we have to do is add the two equations:<\/p>\n

\"\"<\/a><\/p>\n

Once we know x = 4, we can plug into either equation to find that y = \u20132.<\/p>\n

 <\/p>\n

A Slightly More Challenging Example<\/h2>\n

Consider this question:<\/p>\n

2) The symphony sells two kinds of tickets: orchestra, for $40, and upper tiers, for $25.\u00a0 On a certain night, the symphony sells 90 tickets and gets $2625 in revenue from the sales.\u00a0 How many orchestra tickets did they sell?<\/p>\n