Question: What is the equation of the line that passes through the point $(5,8)$ and has a slope of $\frac{5}{3}$?
1. **State the problem:** Find the equation of the line passing through the point $(5,8)$ with slope $\frac{5}{3}$.
2. **Formula used:** The point-slope form of a line is given by
$$y - y_1 = m(x - x_1)$$
where $(x_1,y_1)$ is a point on the line and $m$ is the slope.
3. **Substitute known values:** Here, $x_1 = 5$, $y_1 = 8$, and $m = \frac{5}{3}$. So,
$$y - 8 = \frac{5}{3}(x - 5)$$
4. **Simplify the right side:**
$$y - 8 = \frac{5}{3}x - \frac{5}{3} \times 5 = \frac{5}{3}x - \frac{25}{3}$$
5. **Add 8 to both sides to solve for $y$:**
$$y = \frac{5}{3}x - \frac{25}{3} + 8$$
6. **Convert 8 to a fraction with denominator 3:**
$$8 = \frac{24}{3}$$
7. **Combine the constants:**
$$-\frac{25}{3} + \frac{24}{3} = -\frac{1}{3}$$
8. **Final equation:**
$$y = \frac{5}{3}x - \frac{1}{3}$$
**Answer:** The equation of the line is $y = \frac{5}{3}x - \frac{1}{3}$.