Select the correct answer.

Are the given lines parallel, perpendicular, or neither?

[tex]\[
\begin{array}{l}
3x + 12y = 9 \\
2x - 8y = 4
\end{array}
\][/tex]

A. The slopes of the lines are not the same, so they are perpendicular.
B. The product of the slopes of the lines is 1, so the lines are perpendicular.
C. The slopes of the lines are opposites, so they are neither parallel nor perpendicular.
D. The quotient of the slopes of the lines is 1, so the lines are parallel.



Answer :

Sure! Let's go through a detailed, step-by-step solution:

We have two lines given by the equations:
[tex]\[ \begin{array}{l} 3x + 12y = 9 \\ 2x - 8y = 4 \end{array} \][/tex]

First, let's convert these equations into slope-intercept form, [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope of the line.

1. For the first line [tex]\(3x + 12y = 9\)[/tex]:

[tex]\[ \begin{aligned} 3x + 12y &= 9 \\ 12y &= -3x + 9 \\ y &= \frac{-3}{12}x + \frac{9}{12} \\ y &= -\frac{1}{4}x + \frac{3}{4} \end{aligned} \][/tex]

The slope [tex]\(m_1\)[/tex] of the first line is [tex]\(-\frac{1}{4}\)[/tex].

2. For the second line [tex]\(2x - 8y = 4\)[/tex]:

[tex]\[ \begin{aligned} 2x - 8y &= 4 \\ -8y &= -2x + 4 \\ y &= \frac{2}{8}x - \frac{4}{8} \\ y &= \frac{1}{4}x - \frac{1}{2} \end{aligned} \][/tex]

The slope [tex]\(m_2\)[/tex] of the second line is [tex]\(\frac{1}{4}\)[/tex].

Now, let's analyze the relationship between the two slopes [tex]\(m_1 = -\frac{1}{4}\)[/tex] and [tex]\(m_2 = \frac{1}{4}\)[/tex]:

- The two slopes are [tex]\(-\frac{1}{4}\)[/tex] and [tex]\(\frac{1}{4}\)[/tex].

Checking whether the lines are parallel, perpendicular, or neither involves:

1. Parallel Lines: Two lines are parallel if their slopes are equal, i.e., [tex]\(m_1 = m_2\)[/tex].
[tex]\[ -\frac{1}{4} \ne \frac{1}{4} \][/tex]
Hence, the lines are not parallel.

2. Perpendicular Lines: Two lines are perpendicular if the product of their slopes is [tex]\(-1\)[/tex], i.e., [tex]\(m_1 \cdot m_2 = -1\)[/tex].
[tex]\[ -\frac{1}{4} \times \frac{1}{4} = -\frac{1}{16} \ne -1 \][/tex]
Hence, the lines are not perpendicular.

3. Neither Parallel nor Perpendicular: If the slopes are not equal and their product is not [tex]\(-1\)[/tex], then the lines are neither parallel nor perpendicular.
Since [tex]\(-\frac{1}{4} \neq \frac{1}{4}\)[/tex] and their product [tex]\(-\frac{1}{16} \ne -1\)[/tex], the lines are neither parallel nor perpendicular.

The correct answer is:
- The slopes of the lines are opposites, so they are neither parallel nor perpendicular.

Thus, the solution to the problem is:

[tex]\[ \boxed{3} \][/tex]