Question
Which lines are perpendicular to the line y – 1 = One-third(x+2)? Check all that apply.
y + 2 = –3(x – 4)
y − 5 = 3(x + 11)
y = -3x – Five-thirds
y = One-thirdx – 2
3x + y = 7
y + 2 = –3(x – 4)
y − 5 = 3(x + 11)
y = -3x – Five-thirds
y = One-thirdx – 2
3x + y = 7
Answers
GPT-4o mini
To determine which lines are perpendicular to the line given by the equation \(y - 1 = \frac{1}{3}(x + 2)\), we first need to convert this equation into slope-intercept form (y = mx + b), where m is the slope.
Starting with:
\[
y - 1 = \frac{1}{3}(x + 2)
\]
Distributing \(\frac{1}{3}\):
\[
y - 1 = \frac{1}{3}x + \frac{2}{3}
\]
Adding 1 to both sides:
\[
y = \frac{1}{3}x + \frac{2}{3} + 1
\]
Simplifying:
\[
y = \frac{1}{3}x + \frac{5}{3}
\]
The slope of this line is \(\frac{1}{3}\). The slopes of perpendicular lines are negative reciprocals, so we need the negative reciprocal of \(\frac{1}{3}\), which is \(-3\).
Now, we will check each given line to see if its slope is \(-3\).
1. **Line: \(y + 2 = -3(x - 4)\)**
Converting this to slope-intercept form:
\[
y + 2 = -3x + 12
\]
\[
y = -3x + 10
\]
Slope = \(-3\) (This is perpendicular)
2. **Line: \(y - 5 = 3(x + 11)\)**
Convert to slope-intercept form:
\[
y - 5 = 3x + 33
\]
\[
y = 3x + 38
\]
Slope = \(3\) (Not perpendicular)
3. **Line: \(y = -3x - \frac{5}{3}\)****
This is already in slope-intercept form.
Slope = \(-3\) (This is perpendicular)
4. **Line: \(y = \frac{1}{3}x - 2\)****
This is already in slope-intercept form.
Slope = \(\frac{1}{3}\) (Not perpendicular)
5. **Line: \(3x + y = 7\)**
Convert to slope-intercept form:
\[
y = -3x + 7
\]
Slope = \(-3\) (This is perpendicular)
In conclusion, the lines that are perpendicular to \(y - 1 = \frac{1}{3}(x + 2)\) are:
- \( y + 2 = -3(x - 4) \)
- \( y = -3x - \frac{5}{3} \)
- \( 3x + y = 7 \)
Starting with:
\[
y - 1 = \frac{1}{3}(x + 2)
\]
Distributing \(\frac{1}{3}\):
\[
y - 1 = \frac{1}{3}x + \frac{2}{3}
\]
Adding 1 to both sides:
\[
y = \frac{1}{3}x + \frac{2}{3} + 1
\]
Simplifying:
\[
y = \frac{1}{3}x + \frac{5}{3}
\]
The slope of this line is \(\frac{1}{3}\). The slopes of perpendicular lines are negative reciprocals, so we need the negative reciprocal of \(\frac{1}{3}\), which is \(-3\).
Now, we will check each given line to see if its slope is \(-3\).
1. **Line: \(y + 2 = -3(x - 4)\)**
Converting this to slope-intercept form:
\[
y + 2 = -3x + 12
\]
\[
y = -3x + 10
\]
Slope = \(-3\) (This is perpendicular)
2. **Line: \(y - 5 = 3(x + 11)\)**
Convert to slope-intercept form:
\[
y - 5 = 3x + 33
\]
\[
y = 3x + 38
\]
Slope = \(3\) (Not perpendicular)
3. **Line: \(y = -3x - \frac{5}{3}\)****
This is already in slope-intercept form.
Slope = \(-3\) (This is perpendicular)
4. **Line: \(y = \frac{1}{3}x - 2\)****
This is already in slope-intercept form.
Slope = \(\frac{1}{3}\) (Not perpendicular)
5. **Line: \(3x + y = 7\)**
Convert to slope-intercept form:
\[
y = -3x + 7
\]
Slope = \(-3\) (This is perpendicular)
In conclusion, the lines that are perpendicular to \(y - 1 = \frac{1}{3}(x + 2)\) are:
- \( y + 2 = -3(x - 4) \)
- \( y = -3x - \frac{5}{3} \)
- \( 3x + y = 7 \)