Answer :
Let's analyze each line's equation and identify the slopes first. Then, we'll find the slopes of the lines perpendicular to them. Remember, the slope of the line perpendicular to another line is the negative reciprocal of the original slope.
### 1. Line: [tex]\( y = 9 - \frac{1}{2}(x - 8) \)[/tex]:
First, let's write the given line in the standard slope-intercept form [tex]\(y = mx + b\)[/tex].
[tex]\[ y = 9 - \frac{1}{2}(x - 8) \][/tex]
[tex]\[ y = 9 - \frac{1}{2}x + 4 \][/tex]
[tex]\[ y = -\frac{1}{2}x + 13 \][/tex]
Slope [tex]\( m_1 \)[/tex] of the line [tex]\( y = -\frac{1}{2}x + 13 \)[/tex] is [tex]\(-\frac{1}{2} \)[/tex].
The slope of the perpendicular line will be the negative reciprocal:
[tex]\[ m_{\perp 1} = -\left( -\frac{1}{2} \right)^{-1} = 2 \][/tex].
### 2. Line: [tex]\( y + 4 = -(x + 2) \)[/tex]:
Rewriting in slope-intercept form:
[tex]\[ y + 4 = -(x + 2) \][/tex]
[tex]\[ y + 4 = -x - 2 \][/tex]
[tex]\[ y = -x - 2 - 4 \][/tex]
[tex]\[ y = -x - 6 \][/tex]
Slope [tex]\( m_2 \)[/tex] of the line [tex]\( y = -x - 6 \)[/tex] is [tex]\(-1\)[/tex].
The slope of the perpendicular line will be:
[tex]\[ m_{\perp 2} = -\left( -1 \right)^{-1} = 1 \][/tex].
### 3. Line: [tex]\( y = \frac{1}{3} x - 4 \)[/tex]:
The slope-intercept form is already provided.
Slope [tex]\( m_3 \)[/tex] of the line [tex]\( y = \frac{1}{3} x - 4 \)[/tex] is [tex]\(\frac{1}{3}\)[/tex].
The slope of the perpendicular line will be:
[tex]\[ m_{\perp 3} = -\left( \frac{1}{3} \right)^{-1} = -3 \][/tex].
### 4. Line: [tex]\( y = 3x - 4 \)[/tex]:
The slope-intercept form is already provided.
Slope [tex]\( m_4 \)[/tex] of the line [tex]\( y = 3x - 4 \)[/tex] is [tex]\(3\)[/tex].
The slope of the perpendicular line will be:
[tex]\[ m_{\perp 4} = -\left( 3 \right)^{-1} = -\frac{1}{3} \][/tex].
### Summary:
Now let's match each original line with the corresponding perpendicular line's equation.
1. [tex]\[ y = 9 - \frac{1}{2}(x - 8) \][/tex] has a slope of [tex]\(-\frac{1}{2}\)[/tex]. The perpendicular slope is [tex]\(2\)[/tex].
Thus, it matches with [tex]\( y = 2x + b \)[/tex].
2. [tex]\[ y + 4 = -(x + 2) \][/tex] has a slope of [tex]\(-1\)[/tex]. The perpendicular slope is [tex]\(1\)[/tex].
Thus, it matches with [tex]\( y = x + b \)[/tex].
3. [tex]\[ y = \frac{1}{3}x - 4 \][/tex] has a slope of [tex]\(\frac{1}{3}\)[/tex]. The perpendicular slope is [tex]\(-3\)[/tex].
Thus, it matches with [tex]\( y = -3x + b \)[/tex].
4. [tex]\[ y = 3x - 4 \][/tex] has a slope of [tex]\(3\)[/tex]. The perpendicular slope is [tex]\(-\frac{1}{3}\)[/tex].
Thus, it matches with [tex]\( y = -\frac{1}{3}x + b \)[/tex].
Since we need to match the given equations specifically, let's fix each perpendicular equation accordingly to match the given options:
- [tex]\( y = 2x + b \)[/tex] - does not exactly match any of the given options, but since we don't have another option fitting this requirement, it seems we have some given options in our list which are either wrong representations or we don't have exact perpendicular representations among these options (Yet adjusting into one option remains some visual approximation for [tex]\(y - 9 = 1/2(x -8) \)[/tex] considered approx [tex]\(1/2 \)[/tex]),
- [tex]\( y = x + b \)[/tex] - our comparable line is allowed for visual application,
- [tex]\( y = -3x + b \)[/tex] - matches [tex]\(y-3 x-4\)[/tex]
- [tex]\( y = -\frac{1}{3}x + b \)[/tex] - matches [tex]\( \( y-1 / 3 x-4\)[/tex]
Summarized, we matched some suggested formulae accordingly.
### 1. Line: [tex]\( y = 9 - \frac{1}{2}(x - 8) \)[/tex]:
First, let's write the given line in the standard slope-intercept form [tex]\(y = mx + b\)[/tex].
[tex]\[ y = 9 - \frac{1}{2}(x - 8) \][/tex]
[tex]\[ y = 9 - \frac{1}{2}x + 4 \][/tex]
[tex]\[ y = -\frac{1}{2}x + 13 \][/tex]
Slope [tex]\( m_1 \)[/tex] of the line [tex]\( y = -\frac{1}{2}x + 13 \)[/tex] is [tex]\(-\frac{1}{2} \)[/tex].
The slope of the perpendicular line will be the negative reciprocal:
[tex]\[ m_{\perp 1} = -\left( -\frac{1}{2} \right)^{-1} = 2 \][/tex].
### 2. Line: [tex]\( y + 4 = -(x + 2) \)[/tex]:
Rewriting in slope-intercept form:
[tex]\[ y + 4 = -(x + 2) \][/tex]
[tex]\[ y + 4 = -x - 2 \][/tex]
[tex]\[ y = -x - 2 - 4 \][/tex]
[tex]\[ y = -x - 6 \][/tex]
Slope [tex]\( m_2 \)[/tex] of the line [tex]\( y = -x - 6 \)[/tex] is [tex]\(-1\)[/tex].
The slope of the perpendicular line will be:
[tex]\[ m_{\perp 2} = -\left( -1 \right)^{-1} = 1 \][/tex].
### 3. Line: [tex]\( y = \frac{1}{3} x - 4 \)[/tex]:
The slope-intercept form is already provided.
Slope [tex]\( m_3 \)[/tex] of the line [tex]\( y = \frac{1}{3} x - 4 \)[/tex] is [tex]\(\frac{1}{3}\)[/tex].
The slope of the perpendicular line will be:
[tex]\[ m_{\perp 3} = -\left( \frac{1}{3} \right)^{-1} = -3 \][/tex].
### 4. Line: [tex]\( y = 3x - 4 \)[/tex]:
The slope-intercept form is already provided.
Slope [tex]\( m_4 \)[/tex] of the line [tex]\( y = 3x - 4 \)[/tex] is [tex]\(3\)[/tex].
The slope of the perpendicular line will be:
[tex]\[ m_{\perp 4} = -\left( 3 \right)^{-1} = -\frac{1}{3} \][/tex].
### Summary:
Now let's match each original line with the corresponding perpendicular line's equation.
1. [tex]\[ y = 9 - \frac{1}{2}(x - 8) \][/tex] has a slope of [tex]\(-\frac{1}{2}\)[/tex]. The perpendicular slope is [tex]\(2\)[/tex].
Thus, it matches with [tex]\( y = 2x + b \)[/tex].
2. [tex]\[ y + 4 = -(x + 2) \][/tex] has a slope of [tex]\(-1\)[/tex]. The perpendicular slope is [tex]\(1\)[/tex].
Thus, it matches with [tex]\( y = x + b \)[/tex].
3. [tex]\[ y = \frac{1}{3}x - 4 \][/tex] has a slope of [tex]\(\frac{1}{3}\)[/tex]. The perpendicular slope is [tex]\(-3\)[/tex].
Thus, it matches with [tex]\( y = -3x + b \)[/tex].
4. [tex]\[ y = 3x - 4 \][/tex] has a slope of [tex]\(3\)[/tex]. The perpendicular slope is [tex]\(-\frac{1}{3}\)[/tex].
Thus, it matches with [tex]\( y = -\frac{1}{3}x + b \)[/tex].
Since we need to match the given equations specifically, let's fix each perpendicular equation accordingly to match the given options:
- [tex]\( y = 2x + b \)[/tex] - does not exactly match any of the given options, but since we don't have another option fitting this requirement, it seems we have some given options in our list which are either wrong representations or we don't have exact perpendicular representations among these options (Yet adjusting into one option remains some visual approximation for [tex]\(y - 9 = 1/2(x -8) \)[/tex] considered approx [tex]\(1/2 \)[/tex]),
- [tex]\( y = x + b \)[/tex] - our comparable line is allowed for visual application,
- [tex]\( y = -3x + b \)[/tex] - matches [tex]\(y-3 x-4\)[/tex]
- [tex]\( y = -\frac{1}{3}x + b \)[/tex] - matches [tex]\( \( y-1 / 3 x-4\)[/tex]
Summarized, we matched some suggested formulae accordingly.
