Does (x + y)² + (x – y)² = 170? (1) x² + y² = 85 (2) x > y
A. Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by itself is not.
B. Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by itself is not.
C. Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, even though NEITHER statement BY ITSELF is sufficient.
D. Either statement BY ITSELF is sufficient to answer the question.
E. Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question, meaning that further information would be needed to answer the question.
(A) A hint about working with polynomials on the GMAT: When they are not helpful in one form, change them to another.
In this case, let's expand the binomials on the left side of the equation in the question stem: (x + y)² + (x – y)² = 170 (x² + 2xy + y²) + (x² – 2xy + y²) = 170 2x² + 2y² = 170 2(x² + y²) = 170 x² + y² = 85
Now we can easily see that Statement (1) is sufficient because it gives us the same equation that we simplified the question stem to.
However, Statement (2) is not sufficient by itself because it doesn't provide enough information to solve the question. -------------
I worked out that this is the same equation, but was under the impression that the same equation does not help me here. How does changing the form of the equation answer the question that the initial equation equals 170? Doesn't it just simply show that we are able to simplify the equation, but still have no idea if x and y actually equal 85 and/or 170?
The fastest way to check it is to see that if x² + y² = 85 then (x + y)² + (x – y)² = 170
(x + y)² + (x – y)² = 2(x² + y²) = 2 × 85 = 170
We see that it is so.
Furthermore, let us discuss simplifying of equations:
Even when solving the simplest equation like 2x = 2, we simplify it to x = 1 by dividing each side by 2.
Equation gives us some definite relation between variables in it. The same relation stays for simplified equations or when we do not simplify it but apply actions that lead us to equivalent equation. For example, we have an equation: 2x + 2y= 2
If we simplify it to x + y= 1, we get equivalent equation. If we transform it to 2x + 2y + 2 = 2 + 2, we also get equivalent equation.
However, be careful, because some transformations do not lead to equivalent equations. For example, if we transform 2x + 2y= 2 into 2 + 2y / x = 2 / x, we do not get equivalent equation, because x can not be 0 anymore, though it could in equation 2x + 2y= 2.
If we consider our question, we can see that all the applied transformations lead us to the equivalent equations and therefore equations x² + y² = 85 and (x + y)² + (x – y)² = 170 are equivalent.
Furthermore, note, that solution to our equations is a set of all the pairs (x, y) such that x² + y² = 85. Any pair (x, y) from such set plugged into equations turns them into equalities.
And at the end of our analysis I'd like to mention that this question can become more clear if we consider x,y-plane and areas on it that are defined by each statement.
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