Question: Write the slope-intercept form of the equation of the line through the given points: 15) through: $(1,-5)$ and $(1,-4)$. 16) through: $(0,5)$ and $(1,1)$
1. **Problem 15:** Find the slope-intercept form of the line through points $(1,-5)$ and $(1,-4)$.
2. The slope formula is $$m = \frac{y_2 - y_1}{x_2 - x_1}$$ where $(x_1,y_1) = (1,-5)$ and $(x_2,y_2) = (1,-4)$.
3. Calculate the slope:
$$m = \frac{-4 - (-5)}{1 - 1} = \frac{1}{0}$$ which is undefined.
4. Since the slope is undefined, the line is vertical and its equation is $$x = 1$$.
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5. **Problem 16:** Find the slope-intercept form of the line through points $(0,5)$ and $(1,1)$.
6. Use the slope formula:
$$m = \frac{1 - 5}{1 - 0} = \frac{-4}{1} = -4$$.
7. The slope-intercept form is $$y = mx + b$$.
8. Use point $(0,5)$ to find $b$:
$$5 = -4 \times 0 + b \Rightarrow b = 5$$.
9. Therefore, the equation is:
$$y = -4x + 5$$.
**Final answers:**
- Problem 15: $$x = 1$$ (vertical line)
- Problem 16: $$y = -4x + 5$$