DSQ: Solving equations - GMAT Quantitative

To solve for three unknowns, we need three equations. Here we have three equations if we use both statements 1 and 2. We don't need to solve any further. Because this is a data sufficiency question, it doesn't matter what the actual values of x, y, and z are. The important fact is the we could find them if we wanted to.

z+x=y

Therefore, x=y-z

(1) If z=7, then y-z=y-7, and the value of y-7 can vary. rightarrow NOT sufficient

(2) Subtracting both z and 7 from each side of y+7=z gives y-z=-7.

The value of y-z can be determined. rightarrow SUFFICIENT

We are looking for one value of x since the quesiton specifies we only want a positive solution.

Statement 1 isn't sufficient because there are an infinite number of integers greater than 1.

Statement 2 tells us that x = 2 or x = –2, and we know that we only want the positive answer. Then Statement 2 is sufficient.

We know that we need 2 equations to solve for 2 variables, so it is tempting to say that both statements are needed. This is actually wrong! We aren't being asked for the individual values of x and y, instead we are being asked for the value of an expression.

2x+2y is just 2left ( x+y right ), and statement 2 gives us the value of x+y. For data sufficiency questions, we don't actually have to solve the question, but if we wanted to, we would simply multiply statement 2 by 2.

2x+2y=2left ( x+y right )=2 * Statement 2 = 2times 4=8

In order to solve an equation with 4 variables, you need to know either 3 of the variables or have a system of 4 equations to solve.

To solve for three unknowns, we need three equations. Therefore no combination of statements 1 and 2 will provide enough information to solve for .

You can use the following variation equation to deduce the current volume:

or, equivalently,

To find the current volume , you therefore need three things - the initial volume , which is given in the body of the question; the initial pressure , which you know if you are given Statement 1; and the current pressure, , which you know if you are given Statement 2. Just substitute, and solve.

The first statement is unhelpful; the time of day is irrelevant to the question.

You can use the following variation equation to deduce the current volume:

or, equivalently,

To find the current volume , you therefore need three things - the initial volume , which is given in the body of the question; the current pressure , which you know if you use Statement 2, and the initial pressure , which is not given anywhere.

Let be the current through and the resistance of the flashlight; let be the current through and the resistance of the radio.

The variation equation here would be:

or equivalently:

So in order to find the current in the radio, you need to know three things - the current in the flashlight, which you know from the body of the problem; the resistance from the flashlight , which you know if you are given Statement 2; and the resistance from the radio , which you know if you are given Statement 1. Just substitute and solve.

The difference of two logarithms with the same base is the same-base logarithm of the quotient of the numbers. Therefore, we can simplify this expression as

We need only know the value of , given to us in Statement 1, in order to evaluate this expression.

By the zero product principle, we can solve by setting each linear binomial to zero and solving. This yields three solutions:

Neither statement alone narrows to one of these three solutions, but the two together do.

Solution

To solve for x, we need the value of

Therefore, we need both statements in order to solve for x.

Or

Or

Assume Statement 1 alone.

Assume Statement 2 alone. If , then either or , so we can examine both scenarios.

Case 1:

This is identically false, so we dismiss this case.

Case 2:

Since each equation has only 7 as a solution, either statement alone is sufficient to identify as a true statement.

The absolute value of a positive number is that number; the absolute value of a negative number is the positive opposite.

Assume Statement 1 alone.

Either , in which case , and

or , in which case .

Either or .

Assume Statement 2 alone.

Either , in which case , and

or , in which case .

Either or .

Therefore, neither statement alone proves or disproves that . But if both statements are true, then it must hold that , since this is the only solution of both statements. Therefore, can be proved false.

Assume Statement 1 alone. The solutions of the equation can be found as follows:

The absolute value of a positive number is that number; the absolute value of a negative number is the positive opposite. If , then . If , then also. Therefore, while the value of cannot be determined for certain,either way, is a true statement.

Assume Statement 2 alone. By substitution, we can see two values of that make this true:

, which is true.

, which is also true.

One of these two solutions has absolute value 5, but the other does not. This makes it indefinite whether .

Assume Statement 1 alone. The absolute value of a positive number is that number; the absolute value of a negative number is the positive opposite. As can be seen from substitution, there are two solutions to the equation - and

If , the equation becomes

, a true statement.

If , the equation becomes

, a true statement.

Therefore, that cannot be proved or disproved.

Assume Statement 2 alone.

can be rewritten with the decimal expression in fraction form as

Since if two powers of the same number are equal, the exponents are equal, it follows that .

Assume both statements are true. We show that we cannot determine for certain whether or not is positive.

Case 1:

, satisfying the condition of Statement 1.

The arithmetic mean of 0 and 100 is half their sum, which is , a positive number; the condition of Statement 2 is satisfied.

Case 2:

, satisfying the condition of Statement 1.

The arithmetic mean of 2 and 100 is half their sum, which is , a positive number; the condition of Statement 2 is satisfied.

Therefore, may or may not be positive.

Assume Statement 1 alone. If is a negative number, then of the three given powers of in the set, only (the only even power) is positive. This makes the greatest of the three, thereby giving an answer to the question.

We show that Statement 2 alone is inconclusive by examining two values of whose absolute values are greater than 1 - namely, .

Case 1: . Then and , making the greatest number of the three.

Case 2: . Then and , making the greatest number of the three.

To solve for three unknowns, we need three equations. Here we have three equations if we use both statements 1 and 2. We don't need to solve any further. Because this is a data sufficiency question, it doesn't matter what the actual values of x, y, and z are. The important fact is the we could find them if we wanted to.

z+x=y

Therefore, x=y-z

(1) If z=7, then y-z=y-7, and the value of y-7 can vary. rightarrow NOT sufficient

(2) Subtracting both z and 7 from each side of y+7=z gives y-z=-7.

The value of y-z can be determined. rightarrow SUFFICIENT