Question
What is the equation, in slope-intercept form, of the line that is perpendicular to the given line and passes through the point (2, −1)?
y = Negative one-thirdx − One-third
y = Negative one-thirdx − Five-thirds
y = 3x − 3
y = 3x − 7
y = Negative one-thirdx − One-third
y = Negative one-thirdx − Five-thirds
y = 3x − 3
y = 3x − 7
Answers
GPT-4o mini
To find the equation of a line that is perpendicular to a given line and passes through a specific point, we follow these steps:
1. **Identify the slope of the given line**: The given line is in slope-intercept form \( y = mx + b \). From the provided lines, we can see that the slope \( m \) of the first two equations is \( -\frac{1}{3} \). Therefore, we have:
\( m = -\frac{1}{3} \)
2. **Determine the slope of the perpendicular line**: The slopes of perpendicular lines are negative reciprocals of each other. Thus, if the slope of the given line is \( -\frac{1}{3} \), the slope \( m_{\text{perpendicular}} \) of the perpendicular line will be:
\[
m_{\text{perpendicular}} = -\frac{1}{(-\frac{1}{3})} = 3
\]
3. **Use the point-slope form to write the equation of the perpendicular line**: We now know that the slope of the line we want is \( 3 \) and that it passes through the point \( (2, -1) \). The point-slope form of a line can be expressed as:
\[
y - y_1 = m(x - x_1)
\]
Substituting in the values we have:
- \( m = 3 \)
- \( (x_1, y_1) = (2, -1) \)
We get:
\[
y - (-1) = 3(x - 2)
\]
Simplifying this:
\[
y + 1 = 3(x - 2)
\]
\[
y + 1 = 3x - 6
\]
\[
y = 3x - 6 - 1
\]
\[
y = 3x - 7
\]
Therefore, the equation of the line that is perpendicular to the given line and passes through the point \( (2, -1) \) in slope-intercept form is:
\[
\boxed{y = 3x - 7}
\]
1. **Identify the slope of the given line**: The given line is in slope-intercept form \( y = mx + b \). From the provided lines, we can see that the slope \( m \) of the first two equations is \( -\frac{1}{3} \). Therefore, we have:
\( m = -\frac{1}{3} \)
2. **Determine the slope of the perpendicular line**: The slopes of perpendicular lines are negative reciprocals of each other. Thus, if the slope of the given line is \( -\frac{1}{3} \), the slope \( m_{\text{perpendicular}} \) of the perpendicular line will be:
\[
m_{\text{perpendicular}} = -\frac{1}{(-\frac{1}{3})} = 3
\]
3. **Use the point-slope form to write the equation of the perpendicular line**: We now know that the slope of the line we want is \( 3 \) and that it passes through the point \( (2, -1) \). The point-slope form of a line can be expressed as:
\[
y - y_1 = m(x - x_1)
\]
Substituting in the values we have:
- \( m = 3 \)
- \( (x_1, y_1) = (2, -1) \)
We get:
\[
y - (-1) = 3(x - 2)
\]
Simplifying this:
\[
y + 1 = 3(x - 2)
\]
\[
y + 1 = 3x - 6
\]
\[
y = 3x - 6 - 1
\]
\[
y = 3x - 7
\]
Therefore, the equation of the line that is perpendicular to the given line and passes through the point \( (2, -1) \) in slope-intercept form is:
\[
\boxed{y = 3x - 7}
\]