What is the value of x + y?

Statement 1:

xy + wz + xz + wy = 8
or, x(y+z) + w(y+z) = 8
or, (x+w)(y+z) = 8 . Insufficient

Statement 2:
y + z = 4 . Insufficient

Combining both, we can solve the value of x+y. Hence insufficient

Official Solution:

What is the value of \(x + y\)?

Here we must solve for the value of the expression \(x + y\). We can do so either by solving for the expression as a whole or by solving for \(x\) and \(y\) individually.

Statement 1 says that \(xy + wz + xz + wy = 8\). Whenever we have a chance to factor a polynomial, it is usually a good idea to do so. Note that this polynomial has two terms with a factor of \(x\) and two with a factor of \(w\). Grouping these two terms together: \(8 = xy + wz + xz + wy = (xy + xz) + (wy + wz) = x(y + z) + w(y + z) = (x + w)(y + z)\). This expression is simpler, but without more information, we cannot solve for the value of \(x + y\). Statement 1 does NOT provide sufficient information to answer this question. Eliminate answer choices A and D. The correct answer choice must be B, C, or E.

Statement 2 says that \(y + z = 4\). This gives us no information about either \(x\) or \(w\), and thus we cannot answer the question. Statement 2 does NOT provide sufficient information. Eliminate answer choice B. The correct answer choice is either C or E.

Taken together, statement 1 tells us that \((x + w)(y + z) = 8\), and statement 2 tells us that \(y + z = 4\). Substitute: \((x + w)(4) = 8\). Divide both sides by 4 to get \(x + w = 2\). We now know that \(x + w = 2\) and that \(y + z = 4\). It might be that \(w = 2\), \(x = 0\), \(y = 4\) and \(z = 0\); in that case, \(x + y = 4\). However, it could also be that \(w = 0\), \(x = 2\), \(y = 4\) and \(z = 0\); in that case, \(x + y = 6\). Even together, the statements do NOT provide enough information to find a single value for \(x + y\).

Answer: E.