Solve the equation for x, y and z: $\sqrt{x-y+z}=\sqrt x - \sqrt y + \sqrt z$ I am having some trouble with this problem,

Solve for $x,y,$ and $z$.
  $$\sqrt{x-y+z}=\sqrt x - \sqrt y + \sqrt z$$

Here is my work so far,
$$x - y +z = x+y+z-2\sqrt{xy} + 2\sqrt{xz}- 2\sqrt{zy}$$
$$2y-2\sqrt{xy} + 2\sqrt{xz}- 2\sqrt{zy} = 0 $$
$$2(y-\sqrt{xy} + \sqrt{xz} - \sqrt{zy}) = 0 $$
$$y-\sqrt{xy} + \sqrt{xz} - \sqrt{zy} = 0$$

Even easier, your last equation can be written as:
$$
-\sqrt{y}\left(\sqrt{x} - \sqrt{y}\right) + \sqrt{z}\left(\sqrt{x} - \sqrt{y}\right) = 0
$$
or
$$
\left(\sqrt{z} - \sqrt{y}\right)\left(\sqrt{x} - \sqrt{y}\right) = 0
$$
Which in turn gives you the same conclusion as Calvin pointed out
