What is the value of x - y?

[NandishSS]

What is the value of x - y?

(1) 3x - z = 42 and y - z = 5.

(2) x^2 - xy - xz + yz = 84 and x - z = 12.

OA:B

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Hi NandishSS,

This DS question is all about how you 'combine' pieces of information. To start, it's interesting to note that while the prompt involves 3 variables (X, Y and Z), the question is about the DIFFERENCE of just 2 of the variables (X - Y = ?). As such, to answer the question, there's probably going to be some way to eliminate the third variable...

1) 3X - Z = 42 and Y - Z = 5

If you combine these two equations, you end up with...

3X + Y - 2Z = 47.

If we multiply the second equation by -1, and then combine the two equations we get...

3X - Y = 37

Unfortunately, there's no way to get the value of X - Y from any of this. You can even TEST VALUES and you'll come up with multiple answers for (X - Y).
Fact 1 is INSUFFICIENT

2) X^2 - XY - XZ + YZ = 84 and X - Z = 12

The GMAT is really good at testing you on concepts that you probably know, but in ways that you're not used to thinking about. With Fact 2, what does that first equation remind you of? 4 'pieces' and the first term is a squared-term. That should remind you of a FOIL-ed Quadratic, such as...

(X+1)(X-2) = X^2 - 2X + X - 2

With the given equation, we have all variables though (no numbers), so could you reverse-FOIL that equation...? As a hint, notice that the second equation is X - Z = 12...... Once you complete this step, you'll end up with...

(X-Z)(X-Y) = 84

If you plug in the value for (X-Z), then you'll end up with (12)(X-Y) = 84.... and X-Y = 7
Fact 2 is SUFFICIENT

Final Answer: B

GMAT assassins aren't born, they're made,
Rich

[Brent@GMATPrepNow]

What is the value of x - y?

(1) 3x - z = 42 and y - z = 5.

(2) x² - xy - xz + yz = 84 and x - z = 12.

Target question: What is the value of x - y?

Statement 1: 3x - z = 42 and y - z = 5
Looks like we can eliminate the pesky z's by subtracting the second equation from the first equation.
When we do this, we get: 3x - y = 37
From here, let's TEST some values.
There are several values of x and y that satisfy the equation 3x - y = 37. Here are two:
Case a: x = 13 and y = 2. In this case, x - y = 13 - 2 = 11
Case b: x = 14 and y = 5. In this case, x - y = 14 - 5 = 9
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: x² - xy - xz + yz = 84 and x - z = 12
Notice that we can factor out an x from the first two expressions and factor out a -z from the other two expressions.
We start with: x² - xy - xz + yz = 84
Factor each part to get: x(x - y) -z(x - y) = 84
We get: (x - z)(x - y) = 84
Since we're also told that x - z = 12, we'll replace x - z with 12.
We get: (12)(x - y) = 84
This means that (x - y) = 7
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer: B

Cheers,
Brent

[Jay@ManhattanReview]

What is the value of x - y?

(1) 3x - z = 42 and y - z = 5.

(2) x^2 - xy - xz + yz = 84 and x - z = 12.

OA:B

Hi NandishSS,

It seems that you have a challenge in dealing with statement 2 as statement 1 is pretty simple and leads to insufficiency.

In statement 2, the complicated-looking quadratic equation with three variables needs sorting.

We have x^2 - xy - xz + yz = 84

=> x(x - y) - z(x - y) = 84

=> (x - y)(x - z) = 84

=> (x - y)*12 = 84

=> (x - y) = 7. Sufficient

The correct answer: B

Hope this helps!

Relevant book: Manhattan Review GMAT Number Properties Guide

-Jay
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