Problem Solving

Brent@GMATPrepNow wrote:If xy = 7 and x - y = 7, then x² + y² =
(A) 35
(B) 42
(C) 49
(D) 56
(E) 63

Source: GMAT Prep Now
Difficulty level: 600 - 650

Answer: E

Take x - y = 7; Square both sides to get x^2 - 2xy + y^2 = 49
If xy = 7, we can substitute 7 in place of xy to get x^2 - 2*7 + y^2 = 49
x^2 - 14 + y^2 = 49 ---> x^2 + y^2 = 63. The answer is E

x² + y² =

(x - y)² + 2xy =

7² + 2*7 =

63

We could also look for solutions to x and y, though that's much less fun! Since each term equals 7, we can start by setting them equal to each other.

x - y = xy

Then subtract to get them all on one side:

0 = xy - x + y

Since y = 7/x, we can sub that in:

0 = x*(7/x) - x + (7/x)

Then multiply both sides by x:

0 = 7x - x² + 7

x² - 7x - 7 = 0

Then solve the quadratic, which gives x = 7/2 ± √77/2. Then we know y = -7/2 ± √77/2, since x - y must = 7.

From there, just plug in again, and you're done. Yuck.