GMAT Data Sufficiency questions in the GMAT quantitative (Maths) section need an indepth understanding of how equations work. In the exclusive series for MBA Crystal Ball the GoGMAT experts explain how to tackle Data Sufficiency questions that involve systems of equations.
GMAT Data Sufficiency: Systems of Equations
Most of you are probably familiar with the basic rules for solving linear equations or systems of linear equations, yet many GMAT takers fall prey to booby traps planted in these kinds of Data Sufficiency questions. In order to avoid such mistakes, you should learn well the general methods for solving Systems of Equations and become acquainted with the special cases that may appear on your test.
A set of equations to be solved simultaneously is called a System of Equations. The general rule is that a linear system with n variables usually needs n independent equations to solve it. However, the GMAT Maths questions sometimes uses special cases—offering dependent equations or asking for the value of an expression—that make things a bit more complicated. Let’s examine these more closely.
Beware of DEPENDENT equations
Sometimes an equation in a system does not add essential information but just repeats information already presented by other equations in the system. Such an equation, which is called dependent, is useless and can be eliminated from consideration. Here’s an example:
2x + y + 3z = 1
3x + z = 1
x – y – 2z = 0
If you subtract the first equation from the second, you get:
3x + z – (2x + y + 3z) = 1 – 1
3x + z – 2x – y – 3z = 0
x – y – 2z = 0, the third equation.
This third equation is dependent, because it merely restates the difference between the first two equations. Once this dependent equation is eliminated, the system transforms:
2x + y + 3z = 1
3x + z = 1
Now the number of variables (three) exceeds the number of equations (two), which means that you cannot solve this system to find unique individual values for x, y, and z.
Now look at the following question.
Find the value of xy.
(1) 7 – 2y + 3x = 2
(2) 9x = 6y – 15
A) Statement (1) ALONE is sufficient, but statement (2) is not sufficient.
B) Statement (2) ALONE is sufficient, but statement (1) is not sufficient.
C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D) EACH statement ALONE is sufficient.
E) Statements (1) and (2) TOGETHER are NOT sufficient.
Explanation
To solve for xy, you need only the individual values for x and y. It is clear that neither the first nor the second statement alone is sufficient, as each presents one equation with two variables. (Remember: the number of variables in a linear system is usually the same as the number of independent equations required to solve it.) Thus, you have to choose between Choice E and Choice C. At first glance, the statements together form a system with two variables and two equations, which looks like it could be solved. However, the second equation is the same as the first but multiplied by three: if you rearrange the first equation and multiply by three, you will get the second equation.
7 – 2y + 3x = 2
3x = 2 – 7 + 2y
3x = -5 + 2y, multiply by 3:
9x = 3 (-5 + 2y)
9x = -15 + 6y
9x = 6y – 15
You end up with two equations that are the same, just written differently. Therefore, since you still have only one equation with two variables, you cannot solve it, and the correct answer is Choice E.
Beware of questions that ask you to solve for the VALUE OF AN EXPRESSION
Another common trap is the problem that asks about variables whose number exceeds the number of equations. It is tempting to select answer Choice E, because the system has no single solution for individual variables. However, if the question is about an expression with variables rather than about the variables themselves, it may be possible to define the expression. For example…
Find the value of x + y.
(1) 12 – y^2 – 7 – x^2 = 2xy + 5
(2) (x+y)^4 = 16
A) Statement (1) ALONE is sufficient, but statement (2) is not sufficient.
B) Statement (2) ALONE is sufficient, but statement (1) is not sufficient.
C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D) EACH statement ALONE is sufficient.
E) Statements (1) and (2) TOGETHER are NOT sufficient.
Explanation
Start with the second statement.
(x + y)^4 = 16 ⇔ x + y = [2 or -2]
The value of x + y has two options; therefore, statement (2) alone is insufficient. That eliminates Choice B and Choice D.
The first statement does not immediately appear to be sufficient either, because equations of this type generally have infinitely many pairs of solutions for x and y. However, you don’t have to find values for x and y; you are asked to find the value of x + y. Simplify the equation:
12 – y^2 – 7 – x^2 = 2xy + 5 ⇔ 12 – 12 = x^2 + 2xy + y^2 ⇔ 0 = (x + y)^2
It follows from here that x + y equals 0. Therefore, statement (1) is sufficient and the best answer is Choice A. Note that this equation does not allow you to calculate individual values of x and y, but you can find the expression x + y.
You might face a similar situation in a word problem. For example:
How many hours will it take Ann and Peter, working together at their respective constant rates, to make 100 copies of a certain document?
(1) It takes two hours for Ralph to make 100 copies by himself.
(2) It takes 75 minutes for Ann, Peter, and Ralph working together to make 100 copies.
A) Statement (1) ALONE is sufficient, but statement (2) is not sufficient.
B) Statement (2) ALONE is sufficient, but statement (1) is not sufficient.
C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D) EACH statement ALONE is sufficient.
E) Statements (1) and (2) TOGETHER are NOT sufficient.
Explanation
Let T_A, T_P, T_R and T respectively denote the time (in hours) necessary for making 100 copies when Ann, Peter, Ralph, and Ann & Peter together make copies.
T can be found from the formula:
100/T_A + 100/T_P = 100/T
The first statement yields T_R = 2. It is not sufficient, and you can eliminate Choices A and D.
The second statement says that
100/T_A + 100/T_B + 100/T_R = 100/1.25
This also is not sufficient by itself, and you can eliminate answer Choice B.
To choose between C and E, you must consider the statements together. With three variables and only two equations, you will not able to find values for T_A, T_P and T_R. However, to answer the question, you need only the value of the sum 100/T_A + 100/T_P which you can obtain easily from the available equations.
100/T_A + 100/T_P = 100/1.25 – 100/T_R = 100/1.25 – 100/2
Hence, both statements together are sufficient, and Choice C is the answer.
Sometimes, a GMAT Maths question may not state explicitly the expression you need to calculate. In that case, you take one extra step to find the missing expression.
What is the mean of u+2v and 4u-v?
(1) u + v = 1
(2) 2v + 10u – 7 = 5
A) Statement (1) ALONE is sufficient, but statement (2) is not sufficient.
B) Statement (2) ALONE is sufficient, but statement (1) is not sufficient.
C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D) EACH statement ALONE is sufficient.
E) Statements (1) and (2) TOGETHER are NOT sufficient.
Explanation
First, calculate the expression that you need to evaluate. The mean of u + 2v and 4u – v is equal to (5u + v)/2 and you need to know the values of u and v to find it. That means you must find the combined value of 5u + v and then divide it by two, or find the value of the whole expression (5u + v)/2.
There is a big temptation to answer Choice C, as the two statements together do present a system of two linear equations with two variables, and using the two statements, you can easily find individual values for u and v and calculate the value of the expression. Notice, however, that the second statement contains the doubled expression 5u + v.
Thus, it is sufficient by itself to answer the question; by knowing the value of 2v + 10u – 7, you can derive the value of 5u + v and finally calculate the average.
2v + 10u – 7 = 5 ⇔ 2v + 10u = 12 ⇔ 5u + v = 6 ⇒ (5u + v)/2 = 3
Therefore, the right answer is Choice B.
Even though you want to try to save time on Data Sufficiency questions and have to think abstractly and not solve problems till the end, when you are asked to solve for a variable or an expression, and you see that the two statements present equations, take some extra time to check whether those equations are independent and scan them to see whether either of the equations independently can be manipulated to find the value of the expression. With practice, spotting such opportunities will become intuitive, so practice, pay attention, and accept our best wishes!
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