1. **State the problem:** We need to find the equation of a circle given its center and radius. 2. **Recall the formula for a circle's equation:** The standard form is $$ (x - h)^2 + (y - k)^2 = r^2 $$ where $(h, k)$ is the center and $r$ is the radius. 3. **Identify the given values:** The center is $(5, -5)$ and the radius is $5$ units. 4. **Substitute the values into the formula:** $$ (x - 5)^2 + (y - (-5))^2 = 5^2 $$ 5. **Simplify the equation:** $$ (x - 5)^2 + (y + 5)^2 = 25 $$ 6. **Final answer:** The equation of the circle is $$ (x - 5)^2 + (y + 5)^2 = 25 $$ This equation represents all points $(x, y)$ that are exactly 5 units away from the center $(5, -5)$.