Let's break down how to write the equations in point-slope form for the given pairs of points. The point-slope form of a line's equation is:

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

where:

• $(x_1, y_1)$ is a point on the line.
• $m$ is the slope, calculated as:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

1. Through: (-3,4) and (-4,5)

First, we calculate the slope: $m = \frac{5 - 4}{-4 - (-3)} = \frac{1}{-1} = -1$ Now, using the point (-3, 4) and the slope $m = -1$, the point-slope form is: $y - 4 = -1(x + 3)$

2. Through: (-3,-2) and (0,-4)

Calculate the slope: $m = \frac{-4 - (-2)}{0 - (-3)} = \frac{-2}{3} = -\frac{2}{3}$ Using the point (-3, -2) and the slope $m = -\frac{2}{3}$, the point-slope form is: $y + 2 = -\frac{2}{3}(x + 3)$

3. Through: (0,-5) and (2,3)

Calculate the slope: $m = \frac{3 - (-5)}{2 - 0} = \frac{8}{2} = 4$ Using the point (0, -5) and the slope $m = 4$, the point-slope form is: $y + 5 = 4(x - 0)$ which simplifies to: $y + 5 = 4x$

4. Through: (-3,2) and (4,4)

Calculate the slope: $m = \frac{4 - 2}{4 - (-3)} = \frac{2}{7} = \frac{2}{7}$ Using the point (-3, 2) and the slope $m = \frac{2}{7}$, the point-slope form is: $y - 2 = \frac{2}{7}(x + 3)$

5. Through: (-4,-5) and (-5,-3)

Calculate the slope: $m = \frac{-3 - (-5)}{-5 - (-4)} = \frac{2}{-1} = -2$ Using the point (-4, -5) and the slope $m = -2$, the point-slope form is: $y + 5 = -2(x + 4)$

Let me know if you'd like further clarifications!

Related Questions:

1. How do you convert point-slope form to slope-intercept form?
2. What are the key differences between point-slope form and standard form?
3. How can you find the equation of a line given two points without calculating the slope manually?
4. What does the slope of a line represent geometrically?
5. Can you write an equation of a line parallel to a given line using point-slope form?

Tip:

Always use the point-slope form when given two points or a point and the slope—it is quicker and often easier to handle!