1. **State the problem:** Write the equation of the line in slope-intercept form $y = mt + b$ with slope $m = \frac{a-5}{4}$ passing through the point $(5,0)$. 2. **Recall the slope-intercept form:** The equation of a line is $y = mt + b$, where $m$ is the slope and $b$ is the y-intercept. 3. **Use the point-slope form to find $b$:** Substitute $m = \frac{a-5}{4}$ and the point $(t, y) = (5, 0)$ into $y = mt + b$: $$0 = \frac{a-5}{4} \times 5 + b$$ 4. **Solve for $b$:** $$b = 0 - \frac{5(a-5)}{4} = -\frac{5(a-5)}{4}$$ 5. **Write the final equation:** $$y = \frac{a-5}{4} t - \frac{5(a-5)}{4}$$ This is the equation of the line in slope-intercept form passing through $(5,0)$ with slope $\frac{a-5}{4}$.