<s>[INST] 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?
 [/INST] As the other answers said, your argument already proved there is only one solution.
You can make an independent check by solving the equations graphically.
$(x−1)^2 +(y+1)^2 =25$ is the equation of a circle, radius $5$, center $(1, -1)$. The second equation is another circle, and the circles have two intersection points. The third equation represents a straight line, which goes through just one of the intersection points.
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