1. **State the problem:** We are given the equation of a circle: $$x^2 + y^2 - 14x + 10y + 49 = 0$$ We need to find the coordinates of the center $(x, y)$ and then calculate $x + y$. 2. **Formula and rules:** The general form of a circle's equation is: $$ (x - h)^2 + (y - k)^2 = r^2 $$ where $(h, k)$ is the center and $r$ is the radius. To find the center, we complete the square for both $x$ and $y$ terms. 3. **Complete the square for $x$:** Group $x$ terms: $$x^2 - 14x$$ Take half of $-14$, which is $-7$, and square it: $$(-7)^2 = 49$$ Add and subtract 49 inside the equation to complete the square: $$x^2 - 14x + 49 - 49$$ This becomes: $$(x - 7)^2 - 49$$ 4. **Complete the square for $y$:** Group $y$ terms: $$y^2 + 10y$$ Take half of $10$, which is $5$, and square it: $$5^2 = 25$$ Add and subtract 25 inside the equation: $$y^2 + 10y + 25 - 25$$ This becomes: $$(y + 5)^2 - 25$$ 5. **Rewrite the original equation:** $$x^2 + y^2 - 14x + 10y + 49 = 0$$ Substitute the completed squares: $$ (x - 7)^2 - 49 + (y + 5)^2 - 25 + 49 = 0 $$ Simplify constants: $$ (x - 7)^2 + (y + 5)^2 - 25 = 0 $$ Add 25 to both sides: $$ (x - 7)^2 + (y + 5)^2 = 25 $$ 6. **Identify the center and radius:** Center: $(7, -5)$ Radius: $\sqrt{25} = 5$ 7. **Calculate $x + y$:** $$7 + (-5) = 2$$ **Final answer:** $2$ (Option C)