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To achieve success 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 x?

(1) 2x + 3y = 8

(2) 3x-5y = -7

We have two variables, and once we get both statements, we’ll have two equations, so we’ll be able to solve for x. 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 x?

(1) 3x+4y = 2(x +2y)+3

(2) 4x = y-2

Both equations have two variables, so how could one possibly be sufficient to solve for x? Let’s play with Statement (1) a bit so we can isolate x. Distribute the right side of the equation to get 3x+4y = 2x +4y+3. Then we can subtract 4y from both sides, and poof! We have a single variable equation. We certain can solve for x. 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 2x – y?

(1) 6x + 3y= 15

(2) 6x – 3y = -3

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 2x – y = -1. We still don’t know what x or y is, but we do know what 2x – y is. The answer is (B), statement 2 alone is sufficient.

The Disguised Twin

What is the value of x?

(1) 2x+5y = 12

(2) 4x = 24 – 10y

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 (4x +10y = 24) 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.

If ☉ denotes a mathematical operation, does
x☉y=y☉x for all x and y?

(1) For all x and y, x☉y = 2(x² + y²).
(2) For all y, 0☉y = 2 y²

Highlight to see answer: A

Please post your explanations in the comments below!

If p,q,r, and s are nonzero numbers, is (p – 1)(q – 2)²(r – 3)³(s – 4)4 ≧ 0?

(1) q > 2 and s > 4
(2) p > 1 and r > 3

Highlight to see answers: B

Please post your explanations in the comments below!

Pam and Ed are in a line to purchase tickets. How many people are in the line?

(1) There are 20 people behind Pam and 20 people in front of Ed.
(2) There are 5 people between Pam and Ed.

Highlight to see answer: E

Please post your explanations in the comments below!

If x and y are integers between 10 and 99, inclusive, is (x-y)/9 an integer?

(1) x and y have the same two digits, but in reverse order.
(2) The tens’ digit of x is 2 more than the units digit, and the tens digit of y is 2 less than the units digit.

Highlight to see answer: A

Please post your explanations in the comments below!

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Instead of just aimlessly working on practice problems with no clear insight or direction, this study guide really helped me improve on the different types and techniques that I needed to be focusing on to increase my score.
- Eugene Yu