Answer
$y+4=\frac{3}{4}(x-3)$
Work Step by Step
Step 1. We can identify that the circle $x^2+y^2=25$ is centered at $(0,0)$ with a radius $r=5$
Step 2. We can identify that point $(3,-4)$ is on the circle as $(3)^2+(-4)^2=25$
Step 3. The slope from the center to the point can be found as $m=\frac{-4}{3}=-\frac{4}{3}$
Step 4. As the tangent line is perpendicular to the line above, we have its slope as $n=-\frac{1}{m}=\frac{3}{4}$
Step 5. We can write the equation of the tangent line as $y+4=\frac{3}{4}(x-3)$
