Bunuel wrote:
If x^2 + 2xwy + 3wy^2 = 144, and if x & y & w are all positive, then what is the value (x + 3y) ?
(1) x + 2y = 11
(2) w = 3
Kudos for a correct solution.
MAGOOSH OFFICIAL SOLUTION:We have one equation and three variables. Normally, we would need three equations to solve for the values of the individual variables. Here, though, we are asked to find the value of an expression, and in questions of this sort, often we can find the value of the expression without finding the individual variables. We need (x + 3y)
Statement #1: x + 2y = 11
Well, this gives us the value of another expression, and if we knew the value of y, we could simply add that to this expression to get the value of (x + 3y). The problem is: this would require us to solve for the individual variables, and we have only two equations for three variables. For example, we could solve this for x, x = 11 – 2y, and plug this into the prompt equation to eliminate x, but that would still leave us with one equation with both y and w as unknowns. One equation, two unknowns. We cannot solve for anything with this information. This statement, alone and by itself, is not sufficient.
Statement #2: w = 3
If we plug this into the prompt equation, we get:
\(x^2 + 2x(3)y + 3(3)y^2 = 144\)
\(x^2 + 6xy + 9y^2 = 144\)
This is
the square of a sum, because it fits the pattern:
\(a^2 + 2ab + b^2 = (a + b)^2\)
where a = x, and b = 3y. This means:
\((x + 3y)^2 = 144\).
Normally, when we take a square root, we would have to consider the ± sign. Here, though, we are guaranteed that all three numbers, x & y & w, are positive, so the sum (x + 3y) would have to be positive. Thus, we can take a square root and be sure the answer will be positive.
\(x + 3y = 12\)
This statement lead directly to a numerical answer to the prompt question. This statement, alone and by itself, is sufficient.
Answer = (B)