Given the two equations $|x - z_1| = d_1 + y$ and $|x - z_2| = d_2 + y$ , and suppose that $z_1, z_2 \in \mathbb{R}$, $z_1 \neq z_2$ and $d_1, d_2, \in \mathbb{R}_{> 0}$ are all known reals, solve this system of equations simultaneously for $x \in \mathbb{R}$ and $y \in \mathbb{R}_{\geq 0}$.
First of all, I know that there aren't always solutions. If you take $z_1 = 8$, $z_2 = 11$, $d_1 = 6$ and $d_2 = 1$, it's impossible to find $x$ and $y$ that satify these equations. If you look at the possible value of $x$, by the first equation you get $x = 14 + y$ or $x = 2 - y$. Because we know that $y > 0$, the larger possible solution is $x = 14 + y$. With the same reasoning, we get that the larger solution of the second equation is $x = 12 + y$. There is no $x, y$ that satisfy both equations at once. The same reason applies if you look at the smaller of the two, so $x = 2- y$ and $x = 10 - y$.
Now suppose that there is a solution, how do we find it? Breaking $\mathbb{R}$ up into regions and checking each case where the absolute value changes the sign of the inside expression, is not the way I want to solve this. I want to get general expressions for $x$ and $y$, if that's possible.
I can solve this if $y = 0$ (which I don't know initially, but that's not the point). I square both equations and subract them, so I get a linear equation for $x$, which is easily solved. But I can't use the same trick if $y > 0$. I get the equations $$ x^2 - 2\cdot x\cdot z_1 + z_1^2 = d_1^2 + 2\cdot y\cdot d_1 + y^2 $$ and $$ x^2 - 2\cdot x\cdot z_2 + z_2^2 = d_2^2 + 2\cdot y\cdot d_2 + y^2. $$ I can subtract these, so I get rid of $x^2$ and $y^2$. But then there remains an equation wich has both $x$ and $y$: $$ x\cdot 2(z_2 - z_1) + z_1^2 -z_2^2 = d_1^2 - d_2^2 + y\cdot 2(d_1 -d_2). $$
At this point, I thought about getting rid of $y$ by subtracting the equations, resulting in $$ |x - z_1| - |x-z_2| = d_1 - d_2. $$ If I bring the $|x-z_2|$ to the RHS and squaring this, I get $$ x^2 - 2\cdot x\cdot z_1 + z_1^2 = (d_1-d_2)^2 + 2\cdot (d_1 - d_2)\cdot |x-z_2| + (x^2 - 2\cdot x\cdot z_2 + z_2^2). $$ Now I write this as an equations for $|x-z_2|$: $$ |x-z_2| = \frac{- 2\cdot x\cdot z_1 + z_1^2 - (d_1-d_2)^2 +2\cdot x\cdot z_2 - z_2^2}{2(d_1-d_2)}. $$ It's technically possible to square this equations, resulting in an quadratic expression for $x$. This however, looks very, very unappealing. Another problem is that if I solve this expression, I still don't know an exact value of $x$, because of the $\pm$ in the quadratic formula (I've checked using a problem with known numbers that the root isn't (always) $0$, unfortunately).
So, I'm stuck. I've used all the tricks I known regarding the absolute value, but it doesn't seem to help. I'm now wondering if it is at all possible to get an expression for $x$ and $y$, even if there is a solution.
There are no solutions when $|z_1-z_2|<|d_1-d_2|$, one solution when $|z_1-z_2|>|d_1-d_2|$, and a whole region of solutions when $|z_1-z_2|=|d_1-d_2|$. For the one-solution case, taking $z_2>z_1$, the solution is at $x=(z_1+z_2+d_2-d_1)/2, y=(d_2+d_1+z_2-z_1)/2$.
The argument is pretty much from the geometry of the graphs of these functions.