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

Question

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.