I’m not sure I quite remember how to do these types of problems. What I can offer is 0=0 is the answer I came up with quickly by trying to solve the equation. I couldn’t find my old calc notebook, but you should know what to do when that happens. Or maybe that’s where you’re having trouble too. What section of your math book is this in? That will probably jog my memory.

You wanted step by step.
(uv)/(u+v) = u-(u^2/u+v) multiply both sides by (u+v) from the left side
uv = u^2+uv – u^2 subtraction of u^2
uv = uv divide by uv
0 = 0

Assume u+v != 0.

Multiply u by 1 = (u+v)/(u+v):

(1) u = u(u+v)/(u+v).

Distribute u over u+v in (1):

(2) u = u(u+v)/(u+v) = (u^2+uv)/(u+v).

Subtract u^2/(u+v) from u as expressed in (2):

(3) u-(u^2)/(u+v) = (u^2+uv)/(u+v)-u^2/(u+v).

The fractions in (3) have the same denominator, so simplify the numerator:

(4) (u^2+uv-u^2)/(u+v) = uv/(u+v).

From (3) and (4):

u-(u^2)/(u+v) = uv/(u+v).

@ratboy
Sorry for sounding dumb but.
(2) u = u(u+v)/(u+v) = **(u^2+uv)/(u+v).** WHAT?

so I get for the denominator u+v and u+v is just u+v. but u^2+uv I’m lost on

u(u+v) = uu+uv = u^2+uv.

@ratboy
OMG I’m such a fool thanks so much

Oh actually again

why did you go from (u^2+uv)/(u+v). to subtracting u^2/(u+v) without uv?

u = (u^2+uv)/(u+v), so substitute (u^2+uv)/(u+v) for u in u-(u^2)/(u+v).