<s>[INST] Help with double simultaneous equations and roots I am in the course of a project, in which I need to solve these two simultaneous equations:
\begin{equation}
\sqrt{(1000-y)^2 + x^2} - \sqrt{y^2 + x^2} = 342.371
\end{equation}
\begin{equation}
\sqrt{(2000-x)^2 + y^2} - \sqrt{y^2 + x^2} = 961.674
\end{equation}
I know that the answers are y = 250, and x = 500, but for some reason, I cannot back it up with any calculations, because they always seem to get faulty. I think it has something to do with the square roots, but I was hoping for a useful hand out there.
Just for the sake of the argument, my methodology consists of:
- expanding the formulas
- cancel all equal terms in each formula
- write one variable in terms of another and...
- ... plug in the expression into the second equation.
But my problem already starts with the fact that when I start with the first equation and cancel all equal terms, I am left with an expression containing only one variable. (solving this variable results also in the wrong answer.)
Thanks again for the useful help.
Here my calculations for the first equation, which shall give me an expression to solve the second one:
\begin{equation}
((1000 - y)^2 + x^2) - (y^2 + x^2) = 342.371^2
\end{equation}
\begin{equation}
1000^2 - 2000y + y^2 + x^2 - y^2 - x^2 = 343.371^2
\end{equation}
\begin{equation}
1000^2 - 2000y = 343.371^2
\end{equation}
... and now I am left with an equation without an x in it... and solving this also does not result in the correct solution anyways.
 [/INST] $\sqrt{(1000-y)^2+x^2}-\sqrt{y^2+x^2}=Q$.   
$\sqrt{(1000-y)^2+x^2}=\sqrt{y^2+x^2}+Q$.   
$(1000-y)^2+x^2=y^2+x^2+2Q\sqrt{y^2+x^2}+Q^2$.   
$(1000-y)^2+x^2-y^2-x^2-Q^2=2Q\sqrt{y^2+x^2}$.   
$1000000-2000y-Q^2=2Q\sqrt{y^2+x^2}$.  
Now square both sides to get an equation quadratic in $x$ and $y$. Then go through the same steps with the other equation to get a second quadratic in $x$ and $y$. Then solve the system of 2 equations in 2 unknowns. 
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