Frist one:
x(y + z) = xy + xz. Now, this is equal to xy + z. This means that:
xy + xz = xy + z - subtract xy from each side and you get:
xz = z. Now, shift z to the left side and you have:
xz - z = 0. You can factor z:
z(x - 1) = 0. This product is equal to zero only if one or both factors are equal to zero. So you either have z = 0 or x - 1 = 0, with x = 1.

Second one:
I think that you will agree that 2^x = 2^(x - 2 + 2) = 2^(x - 2) * 2^2.
This means that you can factor 2^(x - 2):
2^x - 2^(x - 2) = [2^(x - 2)][2^2 - 1] = 3*2^(x - 2).
Notice that this is equal to 3*2^13, or that 3*2^(x - 2) = 3*2^13. You can "cut" the 3 from each side and get that 2^(x - 2) = 2^13 or that x - 2 = 13. This makes x = 15.