Answer :
To determine whether each pair of lines is perpendicular, parallel, or neither, we need to analyze the slopes of the lines. Here, the slopes can help us figure out the relationship between the lines.
### Steps to solve this problem:
1. Convert the equations to slope-intercept form (y = mx + b) if necessary, so we can easily identify the slope (m).
2. Identify the slopes of each pair of lines.
3. Compare the slopes:
- Parallel Lines: Two lines are parallel if their slopes are equal.
- Perpendicular Lines: Two lines are perpendicular if the product of their slopes is -1.
- Neither: If the lines are neither parallel nor perpendicular, we classify them as neither.
### Examine each line:
1. Given Line: [tex]\(y = 2x + 4\)[/tex]. The slope is m = 2.
2. Convert [tex]\(2y = 4x - 7\)[/tex] to [tex]\(y = 2x - \frac{7}{2}\)[/tex]. The slope is m = 2.
3. Convert [tex]\(4y = 2x - 4\)[/tex] to [tex]\(y = \frac{1}{2}x - 1\)[/tex]. The slope is m = 0.5.
4. Convert [tex]\(2y = 4x + 4\)[/tex] to [tex]\(y = 2x + 2\)[/tex]. The slope is m = 2.
5. Given Line: [tex]\(y = -2x + 9\)[/tex]. The slope is m = -2.
6. Given Line: [tex]\(y = -2x - 2\)[/tex]. The slope is m = -2.
### Determine Relations:
- Line [tex]\(y = 2x+4\)[/tex] and Line [tex]\(2y = 4x - 7\)[/tex]: Both have slope 2 (Parallel)
- Line [tex]\(y = 2x+4\)[/tex] and Line [tex]\(4y = 2x - 4\)[/tex]: One has slope 2, the other has slope 0.5 (Neither)
- Line [tex]\(y = 2x+4\)[/tex] and Line [tex]\(2y = 4x+4\)[/tex]: Both have slope 2 (Parallel)
- Line [tex]\(y = 2x+4\)[/tex] and Line [tex]\(y = -2x + 9\)[/tex]: One has slope 2, the other has slope -2 (Neither)
- Line [tex]\(y = 2x+4\)[/tex] and Line [tex]\(y = -2x - 2\)[/tex]: One has slope 2, the other has slope -2 (Neither)
Thus, the pairs are:
- Parallel: [tex]\(y = 2x + 4\)[/tex] and [tex]\(2y = 4x - 7\)[/tex], [tex]\(y = 2x + 4\)[/tex] and [tex]\(2y = 4x + 4\)[/tex]
- Neither: [tex]\(y = 2x + 4\)[/tex] and [tex]\(4y = 2x - 4\)[/tex], [tex]\(y = 2x + 4\)[/tex] and [tex]\(y = -2x + 9\)[/tex], [tex]\(y = 2x + 4\)[/tex] and [tex]\(y = -2x - 2\)[/tex]
### Final Answer:
\begin{array}{ccc}
y=2x+4 \quad
\end{array}
\begin{aligned}
\text{Parallel} & \quad \square \\
\text{Perpendicular} & \quad \square \\
\text{Neither} & \quad \square \quad \rightarrow \quad 4 y = 2 x - 4 \quad
\end{aligned}
\begin{array}{ccc}
2y=4x-7 & 2y=4x+4 & y=-2x+9 & y=-2 x - 2\\
\text{Parallel} & \square \quad \rightarrow \quad 2 y = 4 x - 7\\
\text{Perpendicular} & \square\\
\text{Neither} & \square \quad \rightarrow \quad y = -2 x + 9, \quad y = -2 x - 2\\
\end{array}
For clarity:
\begin{array}{ccc}
\text{Parallel:} & 2 y = 4 x - 7, 2 y = 4 x + 4 \\
\text{Neither:} & 4 y = 2 x - 4, y = -2 x + 9, y = -2 x - 2 \\
\text{Perpendicular:} &
\end{array}
Each box corresponds to a pairwise relationship based on the provided analysis.
### Steps to solve this problem:
1. Convert the equations to slope-intercept form (y = mx + b) if necessary, so we can easily identify the slope (m).
2. Identify the slopes of each pair of lines.
3. Compare the slopes:
- Parallel Lines: Two lines are parallel if their slopes are equal.
- Perpendicular Lines: Two lines are perpendicular if the product of their slopes is -1.
- Neither: If the lines are neither parallel nor perpendicular, we classify them as neither.
### Examine each line:
1. Given Line: [tex]\(y = 2x + 4\)[/tex]. The slope is m = 2.
2. Convert [tex]\(2y = 4x - 7\)[/tex] to [tex]\(y = 2x - \frac{7}{2}\)[/tex]. The slope is m = 2.
3. Convert [tex]\(4y = 2x - 4\)[/tex] to [tex]\(y = \frac{1}{2}x - 1\)[/tex]. The slope is m = 0.5.
4. Convert [tex]\(2y = 4x + 4\)[/tex] to [tex]\(y = 2x + 2\)[/tex]. The slope is m = 2.
5. Given Line: [tex]\(y = -2x + 9\)[/tex]. The slope is m = -2.
6. Given Line: [tex]\(y = -2x - 2\)[/tex]. The slope is m = -2.
### Determine Relations:
- Line [tex]\(y = 2x+4\)[/tex] and Line [tex]\(2y = 4x - 7\)[/tex]: Both have slope 2 (Parallel)
- Line [tex]\(y = 2x+4\)[/tex] and Line [tex]\(4y = 2x - 4\)[/tex]: One has slope 2, the other has slope 0.5 (Neither)
- Line [tex]\(y = 2x+4\)[/tex] and Line [tex]\(2y = 4x+4\)[/tex]: Both have slope 2 (Parallel)
- Line [tex]\(y = 2x+4\)[/tex] and Line [tex]\(y = -2x + 9\)[/tex]: One has slope 2, the other has slope -2 (Neither)
- Line [tex]\(y = 2x+4\)[/tex] and Line [tex]\(y = -2x - 2\)[/tex]: One has slope 2, the other has slope -2 (Neither)
Thus, the pairs are:
- Parallel: [tex]\(y = 2x + 4\)[/tex] and [tex]\(2y = 4x - 7\)[/tex], [tex]\(y = 2x + 4\)[/tex] and [tex]\(2y = 4x + 4\)[/tex]
- Neither: [tex]\(y = 2x + 4\)[/tex] and [tex]\(4y = 2x - 4\)[/tex], [tex]\(y = 2x + 4\)[/tex] and [tex]\(y = -2x + 9\)[/tex], [tex]\(y = 2x + 4\)[/tex] and [tex]\(y = -2x - 2\)[/tex]
### Final Answer:
\begin{array}{ccc}
y=2x+4 \quad
\end{array}
\begin{aligned}
\text{Parallel} & \quad \square \\
\text{Perpendicular} & \quad \square \\
\text{Neither} & \quad \square \quad \rightarrow \quad 4 y = 2 x - 4 \quad
\end{aligned}
\begin{array}{ccc}
2y=4x-7 & 2y=4x+4 & y=-2x+9 & y=-2 x - 2\\
\text{Parallel} & \square \quad \rightarrow \quad 2 y = 4 x - 7\\
\text{Perpendicular} & \square\\
\text{Neither} & \square \quad \rightarrow \quad y = -2 x + 9, \quad y = -2 x - 2\\
\end{array}
For clarity:
\begin{array}{ccc}
\text{Parallel:} & 2 y = 4 x - 7, 2 y = 4 x + 4 \\
\text{Neither:} & 4 y = 2 x - 4, y = -2 x + 9, y = -2 x - 2 \\
\text{Perpendicular:} &
\end{array}
Each box corresponds to a pairwise relationship based on the provided analysis.
