A straight line [tex] L_1 [/tex] passes through the points [tex] (0, 3) [/tex] and [tex] (-4, -5) [/tex]. Find:

(a) The slope of the line [tex] L_2 [/tex], which is parallel to [tex] L_1 [/tex].

(b) The slope of the line [tex] L_3 [/tex], which is perpendicular to [tex] L_1 [/tex] and [tex] L_2 [/tex].

Let's go through the steps to find the slopes requested for the question:

1. Slope of Line L1:

The formula to calculate the slope [tex]\( m \)[/tex] of a line passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:

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

For line L1, we have the points [tex]\((0, 3)\)[/tex] and [tex]\((-4, -5)\)[/tex].

- [tex]\( x_1 = 0 \)[/tex], [tex]\( y_1 = 3 \)[/tex]
- [tex]\( x_2 = -4 \)[/tex], [tex]\( y_2 = -5 \)[/tex]

Plug these values into the formula to find the slope of line L1:

[tex]\[
m = \frac{-5 - 3}{-4 - 0} = \frac{-8}{-4} = 2.0
\][/tex]

So, the slope of line L1 is [tex]\( 2.0 \)[/tex].

2. Slope of Line L2:

Line L2 is parallel to line L1. Parallel lines have the same slope. Therefore, the slope of line L2 is the same as the slope of L1:

[tex]\[
\text{slope of L2} = 2.0
\][/tex]

3. Slope of Line L3:

Line L3 is perpendicular to lines L1 and L2. The slope of a line perpendicular to another line is the negative reciprocal of the slope of the original line.

For line L3, with line L1 having a slope of 2.0:

[tex]\[
\text{slope of L3} = -\frac{1}{2.0} = -0.5
\][/tex]

So the slopes are:
- Slope of line L1: [tex]\( 2.0 \)[/tex]
- Slope of line L2: [tex]\( 2.0 \)[/tex] (since it is parallel to L1)
- Slope of line L3: [tex]\( -0.5 \)[/tex] (since it is perpendicular to L1 and L2)