To succeed on the GMAT, there is a general rule of Algebra that you should know: to solve for all variables in a system of equations, you need as many distinct linear equations as variables. So if you get 2 variables, you need two equations; three variables, three equations, and so on. With that in mind, think about this Data Sufficiency question:
What is the value of ?
(1)
(2)
We have two variables, and once we get both statements, we’ll have two equations, so we’ll be able to solve for . The answer is (C), or the third Data Sufficiency answer choice—together the statements are sufficient. If you’ve figured this out, that’s awesome. You’ve discovered how to save a lot of time on Test Day.
But I always tell students not to get trigger-happy. Before you pick (C), keep in mind that the GMAT often gives you situations in which we can get sufficiency with just one equation, or when two won’t be enough. Here are three of those situations:
The Vanishing Variable
What is the value of ?
(1)
(2)
Both equations have two variables, so how could one possibly be sufficient to solve for ? Let’s play with Statement (1) a bit so we can isolate . Distribute the right side of the equation to get . Then we can subtract from both sides, and poof! We have a single variable equation. We certain can solve for . The answer is (A), statement 1 alone is sufficient to answer the question. So before you settle for (C), ask yourself if you can eliminate a variable from one equation.
Solving for a Relationship
What is the value of ?
(1)
(2)
When the GMAT asks you to solve for a relationship between variables (a sum, difference, product, or quotient), ask yourself, Can I manipulate one of the statements to solve for that relationship? If you can do this, you’ll only need one equation for sufficiency. In this case, no amount of manipulating of Statement 1 can do the trick, but let’s play with Statement 2. Divide both sides by 3, and you get . We still don’t know what or is, but we do know what is. The answer is (B), statement 2 alone is sufficient.
The Disguised Twin
What is the value of ?
(1)
(2)
It seems like we have everything we need to pick (C) here. Two equations, two variables, we’re golden. Except dig a little deeper; Statement (2) should cause Déjà vu. Add 10y to both sides of the second equation and divide everything by 2, and you’ll discover that the two equations are identical—just dressed up a little differently. Since we really have just one equation with two variables, we have a recipe for insufficiency. The answer is (E), there is not enough information within these statements to answer the question, no matter how you use them or combine them.
Going forward
So you don’t necessarily have to solve these systems of equations when you see them in a DS question, but you will have to do some detective work before you pick (C). As we advise students in our newly revised GMAT courses, ask yourself the following questions when assessing this type of problem: Can you make a variable vanish? Can you solve for the desired relationship by manipulating an equation? Are these equations really different? When you know the anatomy of the test, you score higher.
Ben Leff
Kaplan GMAT
What about those GMAT DS questions where one linear equation is sufficient by virtue of x and y being multiples of a number?