We are given two points A(1, 3) and B(5, 7). We need to find: (a) The slope of the line passing through A and B. (b) The equation of the line passing through A and B. (c) The equation of the line parallel to the line found in (b) and passing through C(2, -1). (d) The equation of the line perpendicular to the line found in (c) and passing through D(4, 2).

1. Problem Description

We are given two points A(1, 3) and B(5, 7). We need to find:

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

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

(c) The equation of the line parallel to the line found in (b) and passing through C(2, -1).

(d) The equation of the line perpendicular to the line found in (c) and passing through D(4, 2).

2. Solution Steps

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

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

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

In this case, $(x_1, y_1) = (1, 3)$ and $(x_2, y_2) = (5, 7)$.

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

(b) Finding the equation of the line passing through A(1, 3) and B(5, 7).

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 $m = 1$, we have:

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

$y - 3 = x - 1$

$y = x + 2$

(c) Finding the equation of the line parallel to the line found in (b) and passing through C(2, -1).

Since the line is parallel to $y = x + 2$, it has the same slope, $m = 1$.

Using the point-slope form with point C(2, -1):

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

$y + 1 = x - 2$

$y = x - 3$

(d) Finding the equation of the line perpendicular to the line found in (c) and passing through D(4, 2).

The slope of the line in (c) is $m = 1$. The slope of a perpendicular line is the negative reciprocal of $m$. So the slope of the perpendicular line is $m_{\perp} = -\frac{1}{1} = -1$.

Using the point-slope form with point D(4, 2):

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

$y - 2 = -x + 4$

$y = -x + 6$

3. Final Answer

(a) The slope of the line is

1. (b) The equation of the line is $y = x + 2$.

(c) The equation of the parallel line is $y = x - 3$.

(d) The equation of the perpendicular line is $y = -x + 6$.