Explore Westonci.ca, the top Q&A platform where your questions are answered by professionals and enthusiasts alike. Join our platform to connect with experts ready to provide detailed answers to your questions in various areas. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals.
Sagot :
Certainly! Let's solve this step-by-step.
1. Identify the slope of the given path:
The given path is represented by the equation \( y = -4x - 6 \). The slope-intercept form of a linear equation is \( y = mx + b \), where \( m \) is the slope.
Therefore, the slope (\( m_1 \)) of the given path is \( -4 \).
2. Find the slope of the new path:
Since the new path is perpendicular to the given path, its slope (\( m_2 \)) will be the negative reciprocal of the slope of the given path.
[tex]\[ m_2 = -\frac{1}{m_1} = -\frac{1}{-4} = \frac{1}{4} \][/tex]
3. Identify the point of intersection:
The paths intersect at the point \((-4, 10)\).
4. Use the point-slope form of the equation:
The point-slope form of a line passing through a point \((x_1, y_1)\) with slope \( m \) is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Plugging in the slope (\( m_2 = \frac{1}{4} \)) and the point \((-4, 10)\):
[tex]\[ y - 10 = \frac{1}{4}(x + 4) \][/tex]
5. Simplify the equation:
First, distribute the \(\frac{1}{4}\) on the right-hand side:
[tex]\[ y - 10 = \frac{1}{4}x + 1 \][/tex]
Then, add 10 to both sides to solve for \( y \):
[tex]\[ y - 10 + 10 = \frac{1}{4}x + 1 + 10 \][/tex]
[tex]\[ y = \frac{1}{4}x + 11 \][/tex]
So, the equation that represents the new path is:
[tex]\[ \boxed{y = \frac{1}{4}x + 11} \][/tex]
Therefore, the correct answer is D. [tex]\( y = \frac{1}{4}x + 11 \)[/tex].
1. Identify the slope of the given path:
The given path is represented by the equation \( y = -4x - 6 \). The slope-intercept form of a linear equation is \( y = mx + b \), where \( m \) is the slope.
Therefore, the slope (\( m_1 \)) of the given path is \( -4 \).
2. Find the slope of the new path:
Since the new path is perpendicular to the given path, its slope (\( m_2 \)) will be the negative reciprocal of the slope of the given path.
[tex]\[ m_2 = -\frac{1}{m_1} = -\frac{1}{-4} = \frac{1}{4} \][/tex]
3. Identify the point of intersection:
The paths intersect at the point \((-4, 10)\).
4. Use the point-slope form of the equation:
The point-slope form of a line passing through a point \((x_1, y_1)\) with slope \( m \) is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Plugging in the slope (\( m_2 = \frac{1}{4} \)) and the point \((-4, 10)\):
[tex]\[ y - 10 = \frac{1}{4}(x + 4) \][/tex]
5. Simplify the equation:
First, distribute the \(\frac{1}{4}\) on the right-hand side:
[tex]\[ y - 10 = \frac{1}{4}x + 1 \][/tex]
Then, add 10 to both sides to solve for \( y \):
[tex]\[ y - 10 + 10 = \frac{1}{4}x + 1 + 10 \][/tex]
[tex]\[ y = \frac{1}{4}x + 11 \][/tex]
So, the equation that represents the new path is:
[tex]\[ \boxed{y = \frac{1}{4}x + 11} \][/tex]
Therefore, the correct answer is D. [tex]\( y = \frac{1}{4}x + 11 \)[/tex].
We appreciate your time on our site. Don't hesitate to return whenever you have more questions or need further clarification. We hope this was helpful. Please come back whenever you need more information or answers to your queries. We're glad you chose Westonci.ca. Revisit us for updated answers from our knowledgeable team.