Simultaneous Equations in Data Sufficiency

Let’s say that 2x + 8y = 14. What is the value of x? The answer: there’s no way to tell. For any value of x that you choose, there is a value of y that will make the equation true. The equation has an infinite number of solutions.

But what if you knew that 2x + 8y = 14, and you also knew that 5x − 4y = 10? Can we now solve for x? The answer is yes, we can. This illustrates the concept of simultaneous equations, which can be very helpful on the GMAT, particularly with Data Sufficiency questions.

General Rules

A general rule of solving equations is that you need the same number of equations as variables in order to solve for the individual variables themselves. In other words, if you have two variables, you need two equations. Three variables requires three equations, etc. Thus when we talk about simultaneous equations we are saying that these equations are true simultaneously — that they’re all true at the same time. (This same idea is sometimes called working with systems of equations.)

There are several ways of solving simultaneous equations — substitution, elimination with addition and subtraction, graphing — but we aren’t going to focus on how to actually do this. Our focus here is on Data Sufficiency, and one of the nice things about Data Sufficiency is that we don’t have to actually solve anything. As soon as we recognize that the data is sufficient or insufficient our work is done.

Examples

What is the value of x?

1) 3x − y = 20

2) 7x + 5y = 6

You need to determine when the data is sufficient to find the value of x. Fact 1 gives you two variables, but only one equation. This is insufficient to solve for x. Fact 2 also gives you two variables and one equation, so it, likewise, is insufficient. Combining the two statements gives us two variables and two equations, and that is now sufficient to solve for x under the rule for simultaneous equations. Thus the answer is C.

x + y = −12

What is the value of x?

1) 11x + 8y = 22

2) x is not an integer.

Here you have some information in the setup. You know that x + y = −12 before you even turn to the facts. Thus you only need one more equation to solve for x. Fact 1 gives you a second equation, so you now have simultaneous equations which are sufficient. Fact 2 tells you that x is not an integer, but this information is of no help in solving for x. There are an infinite number of non-integers that x could be. Thus the answer is A.

Barbara spends $95 on books. She buys two different kinds of books: novels and comic books. How many novels does Barbara buy?

1) Barbara pays $5 for each novel and $10 for each comic book.

2) Barbara buys a total of 11 books.

This problem is a little different because it doesn’t have explicit equations in it. But equations are lurking beneath the surface, and you want to translate the words into equations. The question wants to know how many novels Barbara buys. Let’s call this x. We’ll call the number of comic books that Barbara buys y. Fact 1 thus tells us that 5x + 10y = 95 because we know from the setup that Barbara spends $95 on books, and Fact 1 tells us the price of each type of book. This is just one equation, however, so it’s insufficient to solve for x. Fact 2 enables us to write the equation x + y = 11. This is also only one equation, however, so it’s insufficient. When we combine the two statements we have simultaneous equations and thus the data is sufficient. The answer is C.

This question illustrates one way the GMAT can make simultaneous equations problems harder — by hiding the equations in a word problem.

A Few Caveats

There are a few things you need to be careful about when applying simultaneous equations. Look at this question.

What is the value of x?

1) 3x + 15y = 35

2) 70 − 6x = 30y

This appears to be a straightforward simultaneous equations problem. Neither fact should be enough by itself, but if you combine them then the data should be sufficient, and the answer should be C. But this question is deceptive. The trick here is that the second equation is really the same as the first. It’s just been doubled and the pieces have been rearranged. If you shuffle the pieces around, the second equation becomes 6x + 30y = 70 which is just the first equation multiplied by 2. Thus the second equation doesn’t tell us anything we didn’t already know, and the answer is E.

So let’s refine the rule for simultaneous equations. With two variables and two distinct equations you can solve for the variables. The equations have to be different. Here’s one more question.

x + 10y = 3

What is the value of x?

1) 5x + 7y = 22

2) − 4y = 21

This question gives us one equation in the setup, so we need a second equation to apply simultaneous equations. Both facts give us a second equation which seems to point to answer D. However, quadratic equations don’t count, so Fact 2 is insufficient. The correct answer is A.

So here’s our final rule: With two variables and two distinct, linear equations (no exponents), you can solve for the variables. Keep an eye out for simultaneous equations in Data Sufficiency questions. They can give you an easy way to determine sufficiency and insufficiency.

Practice a few questions.