I'm currently stuck solving this set of equations. $$x(x+y+z)=4-yz$$ $$y(x+y+z)=9-zx$$ $$z(x+y+z)=25-xy$$
Here's what I've done so far:
By subtracting the second equation from the first, I got
$$(x-y)(x+y+z)=(4-yz)-(9-zx)=-5+z(x-y) \rightarrow (x-y)(x+y)=-5$$
Similarly, by subtracting the third equation from the second, I got
$$(y-z)(x+y+z)=(9-zx)-(25-xy)=-16+x(y-z) \rightarrow (y-z)(y+z)=-16$$
I've also tried adding the three equations together, and got
Unfortunately, I don't think this equation helps.
WolframAlpha tells me that the answer is
but I'm more interested in how the answer is found. Thanks in advance!