Carcass wrote:
If x + y = a and x - y = b, then 2xy = ?
A. \(\frac{(a^2 - b^2)}{2}\)
B. \(\frac{(b^2 - a^2)}{2}\)
C. \(\frac{(a - b )}{2}\)
D. \(\frac{(ab)}{2}\)
E. \(\frac{(a^2 + b^2)}{2}\)
STRATEGY: Upon reading any GRE multiple choice question, we should always ask, Can I use the answer choices to my advantage?
In this case, we can easily test values of x and y.
Now let's give ourselves up to 20 seconds to identify a faster approach.
In this case, we might try some algebraic manipulation, but it's not 100% apparent that this strategy will work. So I'll stick with the strategy I know will work...APPROACH #1: Testing values that satisfy the given information
Since x + y = a and x - y = b, it could be the case that
x = 3 and
y = 2, in which case
2xy = 2(3)(2) = 12 So, we're looking for an answer choice that evaluates to
12Since the answer choices are terms of a and b, we need to find the corresponding values of a and b when
x = 3 and
y = 2So, we'll plug
x = 3 and
y = 2 into the equations x + y = a and x - y = b
Take:
x + y = aSubstitute values to get:
3 + 2 = 5, which means a =
5Take:
x - y = aSubstitute values to get:
3 - 2 = 1, which means b =
1Now plug a =
5 and b =
1 into the 5 answer choices to see which one(s) evaluate to
12...
A. \(\frac{(5^2 - 1^2)}{2} = \)
12. KEEP!
B. \(\frac{(1^2 - 5^2)}{2}=\)
-12. Eliminate
C. \(\frac{(5 - 1 )}{2}=\)
2. Eliminate
D. \(\frac{(5)(1)}{2}=\)
2.5. Eliminate
E. \(\frac{(5^2 + 1^2)}{2}=\)
13. Eliminate
By the process of elimination, the correct answer is A.
APPROACH #2: Algebraic manipulation
Given:
x + y = a
x - y = bAdd the two equations to get:
2x = a + bSimilarly, we can take the two equations....
x + y = a
x - y = b.... And subtract the bottom equation from the top equation to get:
2y = a - bThis means we can write:
(2x)(2y) = (a + b)(a - b)Expand and simplify both sides:
4xy = a² - b²Divide both sides of the equation by 2 to get:
2xy = (a² - b²)/2Answer: A