Question 1065973
1/u + 1/v = 1/f


multiply both sides of the equation by uvf.


you will get uvf/u + uvf/v = uvf/f


simplify to get vf + uf = uv


subtact uf from both sides of the equation to get vf + uf - uf = uv - uf


simplify to get vf = uv - uf


factor out the u on the right hand side of the equation to get vf = u * (v - f)


divide both sides of the equation by (v - f) to get vf / (v - f) = u * (v - f) / (v - f)


simplify to get vf / (v - f) = u


solve for u to get u = vf / (v - f)


i believe that's as simple as it can get.


you can confirm by replacing u with vf / (v - f) in the original equation and evaluating it.


you will get v/vf  = v/vf.


since this is true, the solution looks good.


alternatively, you can give u, v, and f arbitrary values and evaluate both the original equation and the final equation to see if the answers agree.


for example:


u = 2
v = 3


solve this equation and you get f = 6/5.


you get 1/2 + 1/3 = 1/(6/5)


combine like terms on the left and you get 5/6 = 1/(6/5)


multiply both sides of the equation by 6/5 and you get 5/6 * 6/5 = 1/(6/5) * (6/5) which becomes 30/30 = 1 which is true.


now do the same with the final equation, by making u = 2, v = 3, and f = 6/5.


start with u = vf / (v - f)


u = 2
v = 3
f = 6/5
your equation becomes 2 = (3 * 6/5) / (3 - 6/5).


simplify to get 2 = (18/5) / (15/5 - 6/5) which becomes:


2 = (18/5) / (9/5).


this is equivalent to 2 = 18/5 * 5/9.


this can be simplified to 2 = 18/9 which is equal to 2.


since 2 = 2, the equation is true, and the solution is confirmed to be good.