1. **State the problem:** We have a curve defined by the equation $$y = 7x - kx^2$$ where $k$ is a constant. We are told that when $x = 2$, the gradient (derivative) of the curve is $-5$. We need to find the value of $k$. 2. **Recall the formula for the gradient of a curve:** The gradient at any point $x$ is given by the derivative $$\frac{dy}{dx}$$. 3. **Find the derivative of the curve:** $$\frac{dy}{dx} = \frac{d}{dx}(7x - kx^2) = 7 - 2kx$$ 4. **Use the given gradient at $x=2$:** Substitute $x=2$ and gradient $= -5$ into the derivative: $$-5 = 7 - 2k(2)$$ 5. **Solve for $k$:** $$-5 = 7 - 4k$$ Subtract 7 from both sides: $$-5 - 7 = -4k$$ $$-12 = -4k$$ Divide both sides by $-4$: $$\cancel{-12} \div \cancel{-4} = \cancel{-4k} \div \cancel{-4}$$ $$3 = k$$ 6. **Final answer:** The value of $k$ is $3$.