Explore Westonci.ca, the leading Q&A site where experts provide accurate and helpful answers to all your questions. Discover comprehensive answers to your questions from knowledgeable professionals on our user-friendly platform. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals.
Sagot :
To solve this problem, follow these steps:
1. Determine the slope of the original path:
The given equation for the original path is \( y = -3x - 6 \). This equation is in the form \( y = mx + b \), where \( m \) is the slope.
The slope \( m \) of the original path is \( -3 \).
2. Find the slope of the perpendicular path:
For two lines to be perpendicular, the product of their slopes must be \( -1 \). If the slope of the original path is \( -3 \), then let the slope of the perpendicular path be \( m_{\text{perp}} \). We have:
[tex]\[ m \cdot m_{\text{perp}} = -1 \][/tex]
Substituting the slope of the original path:
[tex]\[ -3 \cdot m_{\text{perp}} = -1 \][/tex]
Solving for \( m_{\text{perp}} \):
[tex]\[ m_{\text{perp}} = \frac{1}{-3} = \frac{1}{3} \][/tex]
So, the slope of the perpendicular path is \(\frac{1}{3}\).
3. Use the point-slope form to write the equation of the new path:
The new path must pass through the point of intersection, \((-3, 3)\). Using the point-slope form of the equation of a line, which is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
where \( (x_1, y_1) \) is the point of intersection and \( m \) is the slope of the new path, we can plug in the values:
[tex]\[ y - 3 = \frac{1}{3}(x + 3) \][/tex]
4. Simplify the equation to slope-intercept form \(y = mx + b\):
Distribute \( \frac{1}{3} \) on the right side:
[tex]\[ y - 3 = \frac{1}{3}x + 1 \][/tex]
Add 3 to both sides to isolate \( y \):
[tex]\[ y = \frac{1}{3}x + 4 \][/tex]
So, the equation of the new path is \( y = \frac{1}{3}x + 4 \).
Thus, the correct answer is:
C. [tex]\( y = \frac{1}{3}x + 4 \)[/tex]
1. Determine the slope of the original path:
The given equation for the original path is \( y = -3x - 6 \). This equation is in the form \( y = mx + b \), where \( m \) is the slope.
The slope \( m \) of the original path is \( -3 \).
2. Find the slope of the perpendicular path:
For two lines to be perpendicular, the product of their slopes must be \( -1 \). If the slope of the original path is \( -3 \), then let the slope of the perpendicular path be \( m_{\text{perp}} \). We have:
[tex]\[ m \cdot m_{\text{perp}} = -1 \][/tex]
Substituting the slope of the original path:
[tex]\[ -3 \cdot m_{\text{perp}} = -1 \][/tex]
Solving for \( m_{\text{perp}} \):
[tex]\[ m_{\text{perp}} = \frac{1}{-3} = \frac{1}{3} \][/tex]
So, the slope of the perpendicular path is \(\frac{1}{3}\).
3. Use the point-slope form to write the equation of the new path:
The new path must pass through the point of intersection, \((-3, 3)\). Using the point-slope form of the equation of a line, which is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
where \( (x_1, y_1) \) is the point of intersection and \( m \) is the slope of the new path, we can plug in the values:
[tex]\[ y - 3 = \frac{1}{3}(x + 3) \][/tex]
4. Simplify the equation to slope-intercept form \(y = mx + b\):
Distribute \( \frac{1}{3} \) on the right side:
[tex]\[ y - 3 = \frac{1}{3}x + 1 \][/tex]
Add 3 to both sides to isolate \( y \):
[tex]\[ y = \frac{1}{3}x + 4 \][/tex]
So, the equation of the new path is \( y = \frac{1}{3}x + 4 \).
Thus, the correct answer is:
C. [tex]\( y = \frac{1}{3}x + 4 \)[/tex]
Thank you for visiting our platform. We hope you found the answers you were looking for. Come back anytime you need more information. Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Thank you for visiting Westonci.ca, your go-to source for reliable answers. Come back soon for more expert insights.