This worksheet involves finding the equations of lines using the point-slope form. The point-slope formula is:

$y - y_1 = m(x - x_1)$

where:

• $m$ is the slope,
• $(x_1, y_1)$ is a given point the line passes through.

Here are the solutions to the problems:

1. $m = 2$ through $(3, -5)$

Using the formula: $y - (-5) = 2(x - 3)$ Simplify: $y + 5 = 2(x - 3)$ $y = 2x - 6 - 5$ $y = 2x - 11$

2. $m = -1$ through $(-2, 1)$

Using the formula: $y - 1 = -1(x - (-2))$ Simplify: $y - 1 = -1(x + 2)$ $y = -x - 2 + 1$ $y = -x - 1$

3. $m = \frac{2}{3}$ through $(7, 3)$

Using the formula: $y - 3 = \frac{2}{3}(x - 7)$ Simplify: $y - 3 = \frac{2}{3}x - \frac{14}{3}$ $y = \frac{2}{3}x - \frac{14}{3} + 3$ Convert $3$ to a fraction: $y = \frac{2}{3}x - \frac{14}{3} + \frac{9}{3}$ $y = \frac{2}{3}x - \frac{5}{3}$

4. $m = \frac{1}{4}$ through $(-8, 6)$

Using the formula: $y - 6 = \frac{1}{4}(x - (-8))$ Simplify: $y - 6 = \frac{1}{4}(x + 8)$ $y - 6 = \frac{1}{4}x + 2$ $y = \frac{1}{4}x + 8$

5. Through $(4, 4)$ and $(-2, 8)$

First, find the slope $m$: $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{8 - 4}{-2 - 4} = \frac{4}{-6} = -\frac{2}{3}$ Using $(4, 4)$ and the slope: $y - 4 = -\frac{2}{3}(x - 4)$ Simplify: $y - 4 = -\frac{2}{3}x + \frac{8}{3}$ $y = -\frac{2}{3}x + \frac{8}{3} + 4$ Convert $4$ to a fraction: $y = -\frac{2}{3}x + \frac{8}{3} + \frac{12}{3}$ $y = -\frac{2}{3}x + \frac{20}{3}$

6. Through $(1, 3)$ and $(7, -3)$

Find the slope $m$: $m = \frac{-3 - 3}{7 - 1} = \frac{-6}{6} = -1$ Using $(1, 3)$: $y - 3 = -1(x - 1)$ Simplify: $y - 3 = -x + 1$ $y = -x + 4$

7. Through $(1, -4)$ and $(5, -4)$

Find the slope $m$: $m = \frac{-4 - (-4)}{5 - 1} = \frac{0}{4} = 0$ Since the slope is zero, this is a horizontal line. The equation is: $y = -4$

8. Through $(-4, -2)$ and $(8, -6)$

Find the slope $m$: $m = \frac{-6 - (-2)}{8 - (-4)} = \frac{-6 + 2}{8 + 4} = \frac{-4}{12} = -\frac{1}{3}$ Using $(-4, -2)$: $y - (-2) = -\frac{1}{3}(x - (-4))$ Simplify: $y + 2 = -\frac{1}{3}(x + 4)$ $y + 2 = -\frac{1}{3}x - \frac{4}{3}$ $y = -\frac{1}{3}x - \frac{4}{3} - 2$ Convert $2$ to a fraction: $y = -\frac{1}{3}x - \frac{4}{3} - \frac{6}{3}$ $y = -\frac{1}{3}x - \frac{10}{3}$

Would you like additional steps, clarifications, or details on any specific problem?

Expanded Questions:

1. How can we verify the slope and equation of a line are correct using points on the line?
2. What is the significance of a horizontal line, and how does it differ from vertical lines?
3. How would the equation change if the slope were undefined?
4. Why is it useful to express equations in slope-intercept form after point-slope form?
5. What are some practical applications of writing equations of lines in geometry or science?

Tip: Always double-check your slope calculations to avoid sign errors—these often change the equation entirely!