<s>[INST] Explicit solution to $100(y + \frac{100}{10-y}) = x + 1000$ $$
100\left(y + \frac{100}{10-y}\right) = x + 1000
$$
I'm looking to have the equation in the form $y(x) = ...$ Seems pretty simple, but I'm just not seeing it.
The answer is:
$$
y(x) = \frac{1}{200}\sqrt{x^2 + 4000x} − x
$$
Wolfram|Alpha returns a slightly different answer:
$$
y(x) = \frac{1}{200} (\sqrt{x^2+4000000}+x+2000)
$$
There must be something I'm missing. Thoughts?
 [/INST] Wolfram Alpha is correct. Starting with
$$100\left(y + \frac{100}{10-y}\right) = x + 1000$$
and dividing both sides by $100$:
$$y+\frac{100}{10-y}=\frac{x}{100}+10$$
multiplying both sides by $(10-y)$:
$$-y^2+10y+100=\left(\frac{x}{100}+10\right)(10-y)$$
collecting everything on one side:
$$y^2-\left(20+\frac{x}{100}\right)y+\frac{x}{10}=0$$
and then applying the quadratic formula gives
$$y=\frac{\left(20+\frac{x}{100}\right)\pm\sqrt{\left(20+\frac{x}{100}\right)^2-\frac{2x}{5}}}{2}=\frac{1}{200}\left(2000+x\pm\sqrt{(x+2000)^2-4000x}\right)=$$
$$\frac{1}{200}\left(2000+x\pm\sqrt{x^2+4000000}\right)$$
</s>