How to prove that the following system of equations has only one solution? $
\begin{cases}
(x - 1)^2 + (y + 1)^2 = 25 \\
(x + 5)^2 + (y + 9)^2 = 25 \\
y = -\frac{3}{4}x - \frac{13}{2}
\end{cases}
$
I have to solve this system of equations. After substituting $y = -\frac{3}{4}x - \frac{13}{2}$ into $(x - 1)^2 + (y + 1)^2 = 25$, I got that $(x + 2)^2 = 0$, so $x = -2$, so $y = -\frac{3}{4} \cdot (-2) - \frac{13}{2} = -5$.
How to prove that this is the only solution?

I assume that you have checked that $(x,y) = (-2,-5)$ is indeed a solution.
Why is it the only one?
It is, because you have shown that your three equations must imply that $(x+2)^2=0$, which in turn implies $x=-2$. In other words, whatever solutions $(x,y)$ the three original equations have, you have shown that they must all have the property that $x=-2$.
But given that $x=-2$, the last of the three equations show that we must have $y=-5$. In conclusion, if there is a solution at all to your three equations, then it must be $(x,y)=(-2,-5)$ (and you do need to check that this is a solution, because you can easily end up in the case where there are no solutions at all).
