1. (1,2),(3,-4)   2.  (3,5),  (3,2). 3.   (0,-5),(2,-5). 4.(2,-1),(8,3)

It seems like you're providing pairs of points. Are you looking to find the equations of the lines that pass through these points? Here’s how to approach each pair:

1. **Points (1,2) and (3,-4)**:
   - Slope (m) = \(\frac{y_2 - y_1}{x_2 - x_1} = \frac{-4 - 2}{3 - 1} = \frac{-6}{2} = -3\)
   - Using point-slope form \(y - y_1 = m(x - x_1)\):
     \[
     y - 2 = -3(x - 1) \quad \Rightarrow \quad y = -3x + 5
     \]

2. **Points (3,5) and (3,2)**:
   - This is a vertical line since \(x_1 = x_2 = 3\). The equation is \(x = 3\).

3. **Points (0,-5) and (2,-5)**:
   - This is a horizontal line since \(y_1 = y_2 = -5\). The equation is \(y = -5\).

4. **Points (2,-1) and (8,3)**:
   - Slope (m) = \(\frac{3 - (-1)}{8 - 2} = \frac{4}{6} = \frac{2}{3}\)
   - Using point-slope form:
     \[
     y - (-1) = \frac{2}{3}(x - 2) \quad \Rightarrow \quad y + 1 = \frac{2}{3}x - \frac{4}{3} \quad \Rightarrow \quad y = \frac{2}{3}x - \frac{7}{3}
     \]

### Summary of Line Equations:
1. \(y = -3x + 5\)
2. \(x = 3\) (vertical line)
3. \(y = -5\) (horizontal line)
4. \(y = \frac{2}{3}x - \frac{7}{3}\)

Would you like to explore more details about any of these calculations or concepts? Here are some follow-up questions you might consider:

1. How do you derive the slope from two points?
2. What are the differences between vertical and horizontal lines?
3. Can you provide a real-world example of where these lines might apply?
4. How do you convert from point-slope form to slope-intercept form?
5. What would be the effect on the line's equation if the coordinates of the points were changed?

**Tip**: Always graph the points to visualize the line they create!

Explore how to find equations of lines passing through given points: (1,2) and (3,-4), (3,5) and (3,2), (0,-5) and (2,-5), (2,-1) and (8,3). This solution covers slope calculation, vertical and horizontal lines, and converting to slope-intercept form. Ideal for students in Grades 8-10.

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