Statement (1) can be simplified and looked at conceptually. No need to pick numbers.
(x + z)/2 = y
Since we have no idea what y is, statement (1) is insufficient.
Statement (2) tempts us to use the difference of squares common equation.
(x + y)(x – y) = z
Since x, y and z are all positive, we know x – y must be positive, or x – y > 0. So x > y. But this is about all we can get with this approach, and this is something we could have observed from x^2 – y^2 = z, since z is positive.
The fact that there are no values in x^2 – y^2 = z is a clue that there are many possible combinations of x, y and z that could work. In picking numbers, I always like to start with simple numbers that fit the situation. So since x > y, here are some numbers that fit:
x = 2
y = 1
z = 2^2 – 1 = 3
(x + z)/2 = 2.5
OR
x = 3
y = 2
z = 3^2 – 2^2 = 5
(x + z)/2 = 4
Since we get two possible values for (x + z)/2, statement (2) is insufficient.
For (1) and (2) together, here’s an alternative to picking numbers. Notice that (1) tells us that y is the midpoint of x and z. Since (2) tells us that x > y, we know that z < y < x, and they are all evenly spaced. I like the idea of representing the spacing as d, so y = z + d, and x = z + 2d. Therefore, (2) gives us:
(z + 2d)^2 – (z + d)^2 = z
z^2 + 4dz + 4d^2 – z^2 – 2dz – d^2 = z
2dz + 3d^2 = z
3d^2 = z – 2dz = z(1-2d)
z = 3d^2/(1-2d)
In order to keep z positive, we must have d < ½, but as long as we do that, z could be lots of numbers, and therefore y and x could also be lots of numbers. Since (1) showed that our question is equivalent to y = ?, (1) and (2) together are insufficient.
_________________
SimplyBrilliantPrep.com - Harvard Grad GMAT Instructor with 99th Percentile Scores + Clients Admitted at All Top Business Schools