1. **Problem statement:** We are given that the function $f$ passes through the point $(5,3)$ and that its derivative at $x=5$ is $f'(5)=3$. We want to find the equation of the tangent line to $f$ at $x=5$. 2. **Formula used:** The equation of the tangent line to a function $f$ at a point $x=a$ is given by: $$y = f(a) + f'(a)(x - a)$$ This formula comes from the point-slope form of a line, where the slope is the derivative at $a$ and the line passes through $(a, f(a))$. 3. **Apply the formula:** Here, $a=5$, $f(5)=3$, and $f'(5)=3$. Substitute these values: $$y = 3 + 3(x - 5)$$ 4. **Simplify the equation:** $$y = 3 + 3x - 15 = 3x - 12$$ 5. **Interpretation:** The equation of the tangent line to $f$ at $x=5$ is $y = 3x - 12$. This line touches the graph of $f$ at the point $(5,3)$ and has slope 3 there. **Final answer:** $$y = 3x - 12$$