Let a, b, x, and y be real numbers such that a - b = 1 and y ≠ 0. If the complex number z = x + iy satisfies Im(az + bz + 1) = y, then which of the following is (are) possible value(s) of x?

Im(az + bz + 1) = y
Im((a + b)x + i(a - b)y + 1) = y
Im((a + b)x + iy + 1) = y
(a - b)y = y
Since y ≠ 0, we have a - b = 1, which is given.
Let z = x + iy. Then
az + bz + 1 = a(x + iy) + b(x + iy) + 1 = (a + b)x + i(a - b)y + 1
Im(az + bz + 1) = (a - b)y = y
Given that Im(az + bz + 1) = y, we have (a - b)y = y.
Since y ≠ 0, we can divide by y to get a - b = 1.
Now, let's consider the equation Im(az + bz + 1) = y.
Im(ax + iay + bx + iby + 1) = y
Im((ax + bx + 1) + i(ay + by)) = y
ay + by = y
(a + b)y = y
Since y ≠ 0, we have a + b = 1.
We have the system of equations:
a - b = 1
a + b = 1
Adding the two equations, we get 2a = 2, so a = 1. Then b = 0.
Now, let's use the given condition Im(az + bz + 1) = y:
Im(z + 1) = y
Im(x + iy + 1) = y
y = y
This doesn't help us find x.
Let's go back to Im(az + bz + 1) = y:
Im((a + b)x + i(a - b)y + 1) = y
(a - b)y = y
Since y ≠ 0, a - b = 1 (given).
There must be a mistake in the original problem statement or solution. The given information does not constrain x.