1. **State the problem:** We need to find the value of $b$ such that the gradient (slope) of the line passing through the points $(10,b)$ and $(7,4)$ is 0. 2. **Recall the formula for gradient:** The gradient $m$ between two points $(x_1,y_1)$ and $(x_2,y_2)$ is given by $$m = \frac{y_2 - y_1}{x_2 - x_1}$$ 3. **Apply the formula:** Here, $m=0$, $x_1=10$, $y_1=b$, $x_2=7$, and $y_2=4$. Substitute these values: $$0 = \frac{4 - b}{7 - 10}$$ 4. **Simplify the denominator:** $$0 = \frac{4 - b}{\cancel{7 - 10}^{-3}}$$ 5. **Multiply both sides by $-3$ to eliminate the denominator:** $$0 \times (-3) = 4 - b$$ $$0 = 4 - b$$ 6. **Solve for $b$:** $$b = 4$$ **Final answer:** $b = 4$