1. **State the problem:** We need to find the value of $b$ such that the slope of the line passing through the points $(-2, 3)$ and $(4, b)$ is 12. 2. **Recall the slope formula:** The slope $m$ of a line passing through 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:** Substitute $m=12$, $x_1 = -2$, $y_1 = 3$, $x_2 = 4$, and $y_2 = b$: $$12 = \frac{b - 3}{4 - (-2)} = \frac{b - 3}{4 + 2} = \frac{b - 3}{6}$$ 4. **Solve for $b$:** Multiply both sides by 6: $$12 \times 6 = b - 3$$ $$72 = b - 3$$ Add 3 to both sides: $$72 + 3 = b$$ $$b = 75$$ 5. **Final answer:** The value of $b$ is $75$.