There won’t be that many problems that involve only algebra—maybe 20 percent of the Quantitative section. But a majority of the questions on the Quantitative section will involve algebra in some way. This makes algebra a necessary skill on the GMAT—mastering some frequently tested algebra concepts could yield large improvements on your Quant score. This chapter will guide you through the basics and give you practice applying algebra concepts to GMAT questions ranging from the simple to the most advanced. We'll start by examining the questions we just asked.
For those who find algebra intimidating, this question can seem complicated and abstract at first. Unlike a word problem, which gives you a story to relate to, this question directly measures your ability to use algebraic rules to manipulate unknown terms, or variables. The first step of the Kaplan Method for Problem Solving is to analyze the question stem, and often the best way to start is to “inventory” the question to determine which math skills you will have to use to solve it.
In the question that opens this chapter, you see two algebraic equations, one of which has variables raised to the second power. If you recognized that factoring quadratic equations will be important here, congratulations! If not, don’t worry—you will learn and practice those topics later in this chapter. You are being asked to solve for the algebraic expression x − 2y, so you can anticipate that you will manipulate these two equations to isolate (in other words, put by itself on one side of the equal sign) the expression x − 2y.
This may seem like a somewhat obvious deduction, but it’s a crucial—and potentially reassuring—one: the answer choices here are all numbers, so this means it is possible to solve for a single numerical value for x − 2y. The GMAT can’t ask you a trick question, so you can rest assured that there must be a way to use combination or substitution on these two equations to answer the question. You can think of it like a puzzle, and the test makers have given you all the pieces; it’s up to you to put them together.
A common mistake that test takers make in approaching a question like this one is to think they need to solve for exact values of x and y. While it’s true that values for those two variables would enable you to solve for the expression x − 2y, they are not necessary—in fact, they may be time-consuming or even impossible to solve for. For instance, if x = 8 and y = 3, you would come up with the same value for the expression (x − 2y = 2) as you would if x = −10 and y = −6. An infinite number of values are possible for x and y that would all yield the same result for this expression. Stay focused on the specific task and keep an open mind about how you can use the given information to find the correct answer; often the best path to the solution won’t be the most obvious one.
Here the test maker rewards Pattern Recognition; certain algebraic concepts are tested over and over on the GMAT, and you will learn and practice the most important ones in this chapter. On this question, the test taker who recognizes the opportunity to factor the quadratic equation will answer this question efficiently and accurately. Those test takers who are not as comfortable with the algebraic structures that appear frequently on the GMAT could spend a long time with a question like this one just figuring out where to begin.
Now let’s apply the Kaplan Method for Problem Solving to the algebra question you saw earlier:
Don’t worry if the math is unfamiliar to you; you will have an opportunity to learn it later in the chapter. For now, focus on the basics of applying the Kaplan Method to find the most efficient approach to a GMAT algebra problem.
The GMAT will reward you for recognizing patterns. Here, you are given two variables and two equations, one of which is not linear. When you see an equation that resembles one of the quadratic patterns you know, try factoring it using reverse-FOIL. Odds are the problem will be greatly simplified.
You need to solve for x − 2y. Notice that you are asked for the value of an expression, not an individual variable. Don’t waste time solving for all the variables individually if it’s not necessary.
If you add 10 to both sides of the second equation, you get x2 − xy − 2y2 = 10. Now the left side looks like an expression that you can factor with reverse-FOIL. Since you could get the x2 term by multiplying x and x, you know both factors will contain x:
(x )(x ) = 10
Next, determine what two factors multiplied together will equal −2y2. You need either −2y and y or 2y and −y. Since the coefficient of the xy term is negative, choose −2y and y. Then the sum of the outer and inner products will give you −xy. Thus, the factorization of x2 − xy − 2y2 is:
(x + y)(x − 2y)
Now, notice that both factors of x2 − xy − 2y2 appear in the question stem. You are told that x + y = 2, and you are asked to find the value of x − 2y.
You can find the value of x − 2y by returning to the equation you’ve already factored. You know that (x + y)(x − 2y) = 10. Since x + y = 2, you can replace (x + y) with 2, giving you 2(x − 2y) = 10, or (x − 2y) = 5. So (D) is the correct answer.
Did you get to 2(x − 2y) = 10 and then go further to 2x − 4y = 10 or to x = 5 + 2y? If so, you lost sight of what was asked. You may find it helpful to write down what’s asked on your noteboard so it’s always in your line of sight.
Reread the question stem to check that you have answered the right question. Here, confirm that you have solved for x − 2y and not x or y. When you have done that, you can move on.
Now let’s look at each of the six most important algebra topics that show up on the GMAT Quantitative section, starting with translating words into expressions and equations.