1. **Problem statement:** We are given the function $z = \sqrt{3x + 4y}$ and asked to find the rate of change of $z$ at the point $(3,1)$ as $x$ changes while holding $y$ fixed. 2. **Formula and concept:** The rate of change of $z$ with respect to $x$ holding $y$ constant is the partial derivative $\frac{\partial z}{\partial x}$ evaluated at the point $(3,1)$. 3. **Calculate the partial derivative:** $$z = \sqrt{3x + 4y} = (3x + 4y)^{1/2}$$ Using the chain rule, $$\frac{\partial z}{\partial x} = \frac{1}{2}(3x + 4y)^{-1/2} \cdot 3 = \frac{3}{2\sqrt{3x + 4y}}$$ 4. **Evaluate at the point $(3,1)$:** Calculate the inside of the square root: $$3(3) + 4(1) = 9 + 4 = 13$$ So, $$\frac{\partial z}{\partial x}\bigg|_{(3,1)} = \frac{3}{2\sqrt{13}}$$ 5. **Interpretation:** This value represents how fast $z$ changes as $x$ increases near the point $(3,1)$ when $y$ is held constant. **Final answer:** $$\boxed{\frac{3}{2\sqrt{13}}}$$