Question: A line passes through the points $(3, -8)$ and $(7, -10)$. What is its equation in point-slope form? Use one of the specified points in your equation. Write your answer using integers, proper fractions, and improper fractions. Simplify all fractions.
1. **State the problem:** Find the equation of the line passing through points $(3, -8)$ and $(7, -10)$ in point-slope form.
2. **Formula:** The point-slope form of a line is given by:
$$y - y_1 = m(x - x_1)$$
where $m$ is the slope and $(x_1, y_1)$ is a point on the line.
3. **Calculate the slope $m$:**
$$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-10 - (-8)}{7 - 3} = \frac{-10 + 8}{4} = \frac{-2}{4}$$
4. **Simplify the slope:**
$$m = \frac{\cancel{ -2}}{\cancel{4}} = -\frac{1}{2}$$
5. **Choose one point to use in the equation:** Let's use $(3, -8)$.
6. **Write the point-slope form equation:**
$$y - (-8) = -\frac{1}{2}(x - 3)$$
which simplifies to
$$y + 8 = -\frac{1}{2}(x - 3)$$
**Final answer:**
$$y + 8 = -\frac{1}{2}(x - 3)$$