Bunuel wrote:
The numbers a, b, and c are all positive. If b^2 - c^2 = 17, then what is the value of a^2 - c^2?
(1) a – b = 3
(2) (a + b)/(a - b)= 7
Kudos for a correct solution.
MAGOOSH OFFICIAL SOLUTION:Let X = a^2 - c^2, the value we are seeking. Notice if we subtract the first equation in the prompt from this equation, we get a^2 - b^2 = X – 17. In other words, if we could find the value of a^2 - b^2, then we could find the value of X.
Statement #1: a – b = 3
From this statement alone, we cannot calculate a^2 - b^2, so we can’t find the value of X. Statement #1, alone and by itself, is insufficient.
Statement #2: (a + b)/(a - b) = 7
From this statement alone, we cannot calculate a^2 - b^2, so we can’t find the value of X. Statement #2, alone and by itself, is insufficient.
Statements #1 & #2 combined: Now, if we know both statements are true, then
we could multiple these two equations, which cancel the denominator, and result in the simple equation a + b = 21. Now, we have the numerical value of both (a – b) and (a + b), so from the difference of two squares formula, we can figure out a^2 - b^2, and if we know the numerical value of that, we can calculate X and answer the prompt. Combined, the statements are sufficient.
Answer = C
I don't know why I don't get the highlighted part above. I get the arithmetic behind it but fail to understand why we can do this. I am familiar with systems of equations where you add and subtract equations to eliminate terms. I have a feeling this is very simple but I am just not seeing it. Please help!