DSQ: Solving equations - GMAT Quantitative

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Question

Solve for .

Statement 1:

Statement 2:

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Answer

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

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Question

If $\dpi{100} \small z+x=y$ , what is the value of $\dpi{100} \small x$ ?

(1) $\dpi{100} \small z=7$

(2) $\dpi{100} \small y+7=z$

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Answer

$\dpi{100} \small z+x=y$

Therefore, $\dpi{100} \small x=y-z$

(1) If $\dpi{100} \small z=7$ , then $\dpi{100} \small y-z=y-7$ , and the value of $\dpi{100} \small y-7$ can vary. $\dpi{100} \small \rightarrow$ NOT sufficient

(2) Subtracting both $\dpi{100} \small z$ and 7 from each side of $\dpi{100} \small y+7=z$ gives $\dpi{100} \small y-z=-7$ .

The value of $\dpi{100} \small y-z$ can be determined. $\dpi{100} \small \rightarrow$ SUFFICIENT

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Question

If $\dpi{100} \small x> 0$ , what is the value of x?

Statement 1: $\dpi{100} \small x> 1$

Statement 2: $\dpi{100} \small x^{2}-4=0$

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Answer

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.

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Question

What is the value of $\dpi{100} \small 2x+2y$ ?

Statement 1: $\dpi{100} \small x-3y=4$

Statement 2: $\dpi{100} \small x+y=4$

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Answer

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.

$\dpi{100} \small 2x+2y$ is just $\dpi{100} \small 2\left ( x+y \right )$ , and statement 2 gives us the value of $\dpi{100} \small x+y$ . 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.

$\dpi{100} \small 2x+2y=2\left ( x+y \right )=2$ * Statement 2 = $\dpi{100} \small 2\times 4=8$

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Question

Data Sufficiency Question- do not actually solve the problem

Solve for .

$\small 8vxy+9xz+10vyz=14$

1. $\small x=7$

2. $\small y=2$

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Answer

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.

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Question

Solve for $x,y,and\ z.$

Statement 1:

Statement 2:

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Answer

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.

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Question

The volume of a fixed mass of gas varies inversely with the atmospheric pressure, in millibars, acting upon it, given that all other conditions remain constant.

At 12:00, a balloon was filled with exactly 100 cubic yards of helium. What its current volume?

Statement 1: The atmospheric pressure at 12:00 was 109 millibars.

Statement 2: The atmospheric pressure is now 105 millibars.

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Answer

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

$V_{1} P _{1} = V_{2} P _{2}$

or, equivalently,

$V_{2} = \frac{V_{1} P _{1} }{P _{2}}$

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

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Question

The volume of a fixed mass of gas varies inversely with the atmospheric pressure, in millibars, acting upon it, given that all other conditions remain constant.

At 12:00, a balloon was filled with exactly 100 cubic yards of helium. What is its current volume?

Statement 1: It is now 2:00.

Statement 2: The atmospheric pressure is now 105 millibars.

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Answer

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:

$V_{1} P _{1} = V_{2} P _{2}$

or, equivalently,

$V_{2} = \frac{V_{1} P _{1} }{P _{2}}$

To find the current volume $V_{2}$ , you therefore need three things - the initial volume $V_{1}$ , which is given in the body of the question; the current pressure $P_{2}$ , which you know if you use Statement 2, and the initial pressure $P_{1}$ , which is not given anywhere.

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Question

The electrical current through an object in amperes varies inversely as the object's resistance in ohms, given that all other conditions are equal.

Four batteries hooked up together run an electrical current of 3.2 amperes through John's flashlight. How much current would the same batteries run through John's radio?

Statement 1: The radio has resistance 20 ohms.

Statement 2: The flashlight has resistance 15 ohms.

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Answer

Let $I_{1},R_{1}$ be the current through and the resistance of the flashlight; let $I_{2},R_{2}$ be the current through and the resistance of the radio.

The variation equation here would be:

$\frac{I_{1}}{R_{1}} = \frac{I_{2}}{R_{2}}$

or equivalently:

$\frac{I_{1}}{R_{1}} \cdot R_{2}= \frac{I_{2}}{R_{2}} \cdot R_{2}$

$I_{2} = \frac{I_{1}R_{2}}{R_{1}}$

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

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Question

Evaluate:

$\log_{x} 100y - \log_{x} y$

Statement 1:

Statement 2:

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Answer

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

$\log_{x} 100y - \log_{x} y =\log_{x} \frac{100y}{y} = \log_{x} 100$

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

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Question

is a positive integer

True or false?

Statement 1:

Statement 2: is even.

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Answer

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

$y - 2 = 0 \Rightarrow y = 2$

$y - 4 = 0 \Rightarrow y =4$

$y - 6= 0 \Rightarrow y = 6$

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

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Question

Solve for :

$\left | 4x-3\right |-17=y^{2}-z^{2}$

(1)

(2)

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Answer

Solution

To solve for x, we need the value of $y^{2}-z^{2}$

$y^{2}-z^{2}=(y-z)(y+z)$

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

$\left | 4x-3\right |-17= 4\times12$

$\left | 4x-3\right |-17=48$

$\left | 4x-3\right |=65$

Or

Or

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Question

is a rational number. True or false:

Statement 1: $N^{2} + 4N + 100 = 18N + 51$

Statement 2:

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Answer

Assume Statement 1 alone.

$N^{2} + 4N + 100 = 18N + 51$

$N^{2} + 4N + 100- 18 N - 51 = 18N + 51 - 18 N - 51$

$N^{2} -14N +49 =0$

$(N-7)^{2} = 0$

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.

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Question

True or false:

Statement 1: $X^{2}+ 27 = 12X$

Statement 2: $X^{2}+6X = 27$

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Answer

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.

$X^{2}+ 27 = 12X$

$X^{2}+ 27- 12X= 12X - 12X$

$X^{2}- 12X + 27= 0$

Either , in which case , and

or , in which case .

Either or .

Assume Statement 2 alone.

$X^{2}+6X = 27$

$X^{2}+6X - 27 = 27 - 27$

$X^{2}+6X - 27 = 0$

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.

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Question

True or false:

Statement 1:

Statement 2:

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Answer

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

$N ^{2}+10 N = 10N+25$

$N ^{2}+10 N - 10 N = 10N+25 - 10 N$

$N ^{2}=25$

$N = \pm \sqrt{25} = \pm 5$

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 .

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Question

is a real number. True or false:

Statement 1:

Statement 2: $2^{N} = 0.125$

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Answer

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.

$2^{N} = 0.125$ can be rewritten with the decimal expression in fraction form as

$2^{N} = \frac{1}{8}$

$2^{N} = \frac{1}{2^{3}}$

$2^{N} = 2^{-3}$

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

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Question

is a real number. True or false: is positive.

Statement 1: $3^{N} > \frac{1}{3}$

Statement 2: The arithmetic mean of 100 and is positive.

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Answer

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

Case 1:

$3^{N} = 3^{0} = 1 > \frac{1}{3}$ , satisfying the condition of Statement 1.

The arithmetic mean of 0 and 100 is half their sum, which is $\frac{0+100}{2} = 50$ , a positive number; the condition of Statement 2 is satisfied.

Case 2:

$3^{N} = 3^{2} = 9 > \frac{1}{3}$ , satisfying the condition of Statement 1.

The arithmetic mean of 2 and 100 is half their sum, which is $\frac{2+100}{2} = 51$ , a positive number; the condition of Statement 2 is satisfied.

Therefore, may or may not be positive.

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Question

Solve for $x,y,and\ z.$

Statement 1:

Statement 2:

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Answer

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.

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Question

If $\dpi{100} \small z+x=y$ , what is the value of $\dpi{100} \small x$ ?

(1) $\dpi{100} \small z=7$

(2) $\dpi{100} \small y+7=z$

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Answer

$\dpi{100} \small z+x=y$

Therefore, $\dpi{100} \small x=y-z$

(1) If $\dpi{100} \small z=7$ , then $\dpi{100} \small y-z=y-7$ , and the value of $\dpi{100} \small y-7$ can vary. $\dpi{100} \small \rightarrow$ NOT sufficient

(2) Subtracting both $\dpi{100} \small z$ and 7 from each side of $\dpi{100} \small y+7=z$ gives $\dpi{100} \small y-z=-7$ .

The value of $\dpi{100} \small y-z$ can be determined. $\dpi{100} \small \rightarrow$ SUFFICIENT

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Question

If $\dpi{100} \small x> 0$ , what is the value of x?

Statement 1: $\dpi{100} \small x> 1$

Statement 2: $\dpi{100} \small x^{2}-4=0$

Tap to reveal answer

Answer

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.

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