What is the value of x - y?

What is the value of x - y?

(1) y = 7 - x
Don't know value of y - insufficient

(2) 2 √xy=7
xy=49/4=12.25 - Insufficient

1+2 -> Substitute value of y=(7-x) in place of y^2 in below equation.

(x-y)^2 = x^2 + y^2 - 2xy = x^2 + (7-x)^2 - 2*12.25=0 --> x^2 - 7x + 12.25=0

Solving the above equation we get x = 3.5, 3.5. So y=3.5.

x-y = 0. Ans is C.

Thanks,

Please give me the Kudos.

our question is to find value of x-y = ?

1) y = 7-x can’t compute value of x-y from this so insufficient

2) 2√xy=7 can’t compute value of x-y from this so insufficient

1) + 2) solving both we get value of x and y and hence we can compute value of x-y

Hence C is the correct answer

Question : x-y = ?

Statement 1: y = 7 - x

i.e. x+y = 7 which has multiple solutions for (x,y) e.g. (4, 3) or (5, 2)

Therefore, x-y can have multiple solution

Hence, NOT SUFFICIENT

Statement 2: \(2\sqrt{xy}=7\)

Squaring both sides
i.e. \(4*x*y = 49\)

i.e. \(x*y = (7/2)^2 = 49/4\)

But this may have multiple solution for (x, y) e.g. (7/2, 7/2) or (49/4, 1)

Hence, NOT SUFFICIENT

Combining the two statements

\((x-y)^2 = (x+y)^2 - 4xy\)

i.e. \((x-y)^2 = (7)^2 - 49\)

i.e. \((x-y)^2 =49 - 49 = 0\)

Hence, SUFFICIENT

Answer: Option

MANHATTAN GMAT OFFICIAL SOLUTION:

(1) INSUFFICIENT: This tells us that x + y = 7, but nothing about the value of x – y.

(2) INSUFFICIENT: We cannot manipulate this statement into the form x – y, nor can we determine the value of x or y independently.

(1) AND (2) SUFFICIENT: First, recognize that xy must be positive in order \(\sqrt{xy}\) for in statement (2) to be a real number, so x and y must have the same sign. In order for x + y in statement (1) to be a positive number (i.e. 7), the sign of x and y must be positive, because negative + negative cannot be positive. So in the following steps, we know that the square roots of x and y are positive, real numbers.

From the statements, we know that x + y = 7 and \(2\sqrt{xy}=7\). Since both equal 7, we can set the left sides of the equations equal and simplify:
\(x+y=2\sqrt{xy}\)
\(x-2\sqrt{xy}+y=0\)

Recognizing the “square of a difference” special product, we can write this in factored form:
\((\sqrt{x}-\sqrt{y})^2=0\)

Thus, \(\sqrt{x}-\sqrt{y}=0\), so \(\sqrt{x}=\sqrt{y}\). We can infer from this that x = y, and x – y = 0.

The correct answer is C.

Bunuel ,

Why can't we get from 1) x-(7-x)=? 2x=7 x=3.5 substitute into the original and get y=7-3.5 y=3.5 and get an answer?...

We need the value of x-(7-x)=2x-7. It does NOT mean that 2x=7.

Bunuel,

Thank you for your reply!

So does this mean that the 2x-7=? could be 2x-7=100 or 2x-7= 150^34 and we would get different values for x? So we need more information to get to an exact value?

Pff this is really confusing for me because I've always assumed that when you have an equation with one variable, you can solve for that variable..

Or we don't have an equation with one variable here? We have like "2x-7=y determine y" ?

Thank you!

y = 7 - x is a linear equation with TWO variables, x and y.

mikemcgarry, help please!

Why not substitute value of y in the given equation and get:

x-(7-x)=?
2x=7
x=3.5

since y=7-x we get the y value and get the solution!

Dear iliavko,

I'm happy to respond. My friend, you set an expression to an unknown and promptly set that unknown equal to zero.

Your first step: x - (7 - x) = Q
Rather than use a question mark, I am going to use a letter, Q, for the unknown. Letters for unknowns is a standard algebra move, with good reason.

Your second step should be: 2x = 7 + Q
We made the dubious choice of introducing a new unknown into the problem. This route will not lead to an answer.

I recommend avoiding question marks as algebraic symbols, because all punctuation has a tenuous relationship with math. Instead, use a letter, because then our tried & true algebraic instincts kick in.

Does all this make sense?
Mike

mikemcgarry

Thank you SO much for your explanation!! Yes it does make sense now!

Interesting that I just assumed that the expression was =0 and didn't even notice!

Your explanations are just great!

Thanks a lot, cheers!