Let's determine the relationship between each pair of lines.
### 1. First Pair: [tex]\(2y = 4x + 4\)[/tex] and [tex]\(y = -2x - 2\)[/tex]
To identify the relationship between these lines, we need to express both equations in the slope-intercept form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope.
- For the equation [tex]\(2y = 4x + 4\)[/tex]:
[tex]\[
y = \frac{4x + 4}{2} \\
y = 2x + 2
\][/tex]
So, the slope [tex]\(m_1 = 2\)[/tex].
- For the equation [tex]\(y = -2x - 2\)[/tex]:
[tex]\[
y = -2x - 2 \ (\text{already in slope-intercept form})
\][/tex]
So, the slope [tex]\(m_3 = -2\)[/tex].
Two lines are perpendicular if the product of their slopes is [tex]\(-1\)[/tex]. Here:
[tex]\[
2 \times -2 = -4
\][/tex]
Thus, these lines are Perpendicular.
### 2. Second Pair: [tex]\(y = -2x - 2\)[/tex] and [tex]\(4y = 2x - 4\)[/tex]
Let's convert both lines to the slope-intercept form.
- For the equation [tex]\(4y = 2x - 4\)[/tex]:
[tex]\[
y = \frac{2x - 4}{4} \\
y = \frac{1}{2}x - 1
\][/tex]
So, the slope is [tex]\(m_2 = \frac{1}{2}\)[/tex].
- For [tex]\(y = -2x - 2\)[/tex]:
[tex]\[
y = -2x - 2 \ (\text{already in slope-intercept form})
\][/tex]
So, the slope [tex]\(m_3 = -2\)[/tex].
The slopes [tex]\(-2\)[/tex] and [tex]\(\frac{1}{2}\)[/tex] do not multiply to [tex]\(-1\)[/tex] and they are not equal, hence these lines are Neither parallel nor perpendicular.
### 3. Third Pair: [tex]\(4y = 2x - 4\)[/tex] and [tex]\(y = -2x + 9\)[/tex]
Express both equations in the slope-intercept form.
- For the equation [tex]\(4y = 2x - 4\)[/tex]:
[tex]\[
y = \frac{2x - 4}{4} \\
y = \frac{1}{2}x - 1
\][/tex]
So, the slope is [tex]\(m_2 = \frac{1}{2}\)[/tex].
- For [tex]\(y = -2x + 9\)[/tex]:
[tex]\[
y = -2x + 9 \ (\text{already in slope-intercept form})
\][/tex]
So, the slope [tex]\(m_4 = -2\)[/tex].
The slopes [tex]\(\frac{1}{2}\)[/tex] and [tex]\(-2\)[/tex] do not multiply to [tex]\(-1\)[/tex] and they are not equal, hence these lines are Neither parallel nor perpendicular.
### 4. Fourth Pair: [tex]\(y = -2x + 9\)[/tex] and [tex]\(y = 2x + 4\)[/tex]
Express both equations in the slope-intercept form.
- For [tex]\(y = -2x + 9\)[/tex]:
[tex]\[
y = -2x + 9 \ (\text{already in slope-intercept form})
\][/tex]
So, the slope [tex]\(m_4 = -2\)[/tex].
- For [tex]\(y = 2x + 4\)[/tex]:
[tex]\[
y = 2x + 4 \ (\text{already in slope-intercept form})
\][/tex]
So, the slope [tex]\(m_5 = 2\)[/tex].
The slopes [tex]\(-2\)[/tex] and [tex]\(2\)[/tex] do not multiply to [tex]\(-1\)[/tex] and they are not equal, hence these lines are Neither parallel nor perpendicular.
Now, let's place the correct relationships in the correct boxes:
- Neither: [tex]\(\square\)[/tex]
- [tex]\(y = -2x - 2\)[/tex] and [tex]\(4y = 2x - 4\)[/tex]
- [tex]\(4y = 2x - 4\)[/tex] and [tex]\(y = -2x + 9\)[/tex]
- [tex]\(y = -2x + 9\)[/tex] and [tex]\(y = 2x + 4\)[/tex]
- Perpendicular: [tex]\(\square\)[/tex]
- [tex]\(2y = 4x + 4\)[/tex] and [tex]\(y = -2x - 2\)[/tex]
So, we have:
Neither:
- [tex]\(y = -2x - 2\)[/tex] and [tex]\(4y = 2x - 4\)[/tex]
- [tex]\(4y = 2x - 4\)[/tex] and [tex]\(y = -2x + 9\)[/tex]
- [tex]\(y = -2x + 9\)[/tex] and [tex]\(y = 2x + 4\)[/tex]
Perpendicular:
- [tex]\(2y = 4x + 4\)[/tex] and [tex]\(y = -2x - 2\)[/tex]
