Adding and Removing Non-Compounded Percentages does not produce the same result?

Let's take a simpler example first. If you increase 400 by 25% you get 500. If you reduce 500 by 25% you get 375. With equations this is $$400 \times (1+ 0.25) = 500$$ $$500 \times (1- 0.25) = 375$$ so it is clear that this multiplication does not work, and we should not expect it to since $(1+ 0.25) \times (1- 0.25) = 0.9375$ not $1$. So to undo the increase (from 400 to 500), instead we should divide as follows: $$500 \div (1+ 0.25) = 400$$ and since $1 \div (1+ 0.25) = (1 - 0.20) $, we would have $$500 \times (1 - 0.20) = 400$$ so a 25% increase must be undone by a 20% decrease. You seem to understand this in your equation for value.

The second issue is your lack of compounding the taxes. If you did compound then it is correct that $$100 \times (1 + 0.10) \times (1 + 0.07) = 117.70$$ and you would undo it with $$117.70 \div (1 + 0.10) \div (1 + 0.07) = 100$$ but that does not work here.

So instead you start with $$100 \times (1 +0.10 + 0.07) = 117$$ and to undo it you do $$117 \div (1 +0.10 + 0.07) = 100$$

Essentially, you've computed your after-tax value using the formula $$\array{T &=& P + (0.10)P + (0.07)P\\&=& (1.17)P}$$ where T is the after-tax price, and P is the before-tax price.

If you want to get back to P from T, just solve for P in the above equation to get $$P = \frac{T}{1.17}$$