The problem asks us to find the slope and equation of a line passing through two given points A(1, 3) and B(5, 7). Then, we need to find the equation of a line parallel to the first line and passing through point C(2, -1). Finally, we need to find the equation of a line perpendicular to the second line and passing through point D(4, 2).

1. Problem Description

The problem asks us to find the slope and equation of a line passing through two given points A(1, 3) and B(5, 7). Then, we need to find the equation of a line parallel to the first line and passing through point C(2, -1). Finally, we need to find the equation of a line perpendicular to the second line and passing through point D(4, 2).

2. Solution Steps

(a) Find the slope of the line passing through points A(1, 3) and B(5, 7).

The formula for the slope $m$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

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

Using A(1, 3) and B(5, 7):

$m = \frac{7 - 3}{5 - 1} = \frac{4}{4} = 1$

(b) Find the equation of the line passing through points A and B.

We can use the point-slope form of a line:

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

Using point A(1, 3) and the slope $m = 1$:

$y - 3 = 1(x - 1)$

$y - 3 = x - 1$

$y = x + 2$

(c) Find the equation of the line that is parallel to the line found in (b) and passes through the point C(2, -1).

Parallel lines have the same slope. The line in (b) has a slope of

1. So, the parallel line also has a slope of

1. Using the point-slope form with point C(2, -1) and $m = 1$:

$y - (-1) = 1(x - 2)$

$y + 1 = x - 2$

$y = x - 3$

(d) Find the equation of the line that is perpendicular to the line found in (c) and passes through the point D(4, 2).

The slope of the line in (c) is

1. The slope of a line perpendicular to this line is the negative reciprocal of 1, which is -

1. So, the perpendicular line has a slope of -

1. Using the point-slope form with point D(4, 2) and $m = -1$:

$y - 2 = -1(x - 4)$

$y - 2 = -x + 4$

$y = -x + 6$

3. Final Answer

(a) The slope of the line passing through points A and B is

1. (b) The equation of the line passing through points A and B is $y = x + 2$.

(c) The equation of the line parallel to the line in (b) and passing through C(2, -1) is $y = x - 3$.

(d) The equation of the line perpendicular to the line in (c) and passing through D(4, 2) is $y = -x + 6$.