Question

1. The line passing through the points (2,-5) and (6,2).
2. The line passing through the points (4,-3) and (8,4).

Find the slopes of the lines that passes through the given points. Then determine if the two lines are parallel or perpendicular or neither.

Answer 1

To find the slopes of the lines passing through the given points, we will use the formula for the slope \( m \) of a line that passes through two points \((x_1, y_1)\) and \((x_2, y_2)\):

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

### 1. Finding the slope of the line through points (2, -5) and (6, 2):

Here, \((x_1, y_1) = (2, -5)\) and \((x_2, y_2) = (6, 2)\).

Using the slope formula:

\[
m_1 = \frac{2 - (-5)}{6 - 2} = \frac{2 + 5}{6 - 2} = \frac{7}{4}
\]

### 2. Finding the slope of the line through points (4, -3) and (8, 4):

Here, \((x_1, y_1) = (4, -3)\) and \((x_2, y_2) = (8, 4)\).

Using the slope formula:

\[
m_2 = \frac{4 - (-3)}{8 - 4} = \frac{4 + 3}{8 - 4} = \frac{7}{4}
\]

### Comparison of the slopes:

- The slope of the first line is \( m_1 = \frac{7}{4} \).
- The slope of the second line is \( m_2 = \frac{7}{4} \).

Since both lines have the same slope (\( m_1 = m_2 \)), we can conclude that the two lines are **parallel**.

### Summary:
- Slope of the first line: \( \frac{7}{4} \)
- Slope of the second line: \( \frac{7}{4} \)
- Conclusion: The lines are parallel.