Get reliable answers to your questions at Westonci.ca, where our knowledgeable community is always ready to help. Join our platform to connect with experts ready to provide precise answers to your questions in different areas. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.
Sagot :
To solve for the equation of the line that is perpendicular to [tex]\( y = -\frac{1}{2} x - 5 \)[/tex] and passes through the point [tex]\( (2, 7) \)[/tex], we need to follow these steps:
1. Identify the slope of the given line:
The equation of the given line is [tex]\( y = -\frac{1}{2} x - 5 \)[/tex]. From this equation, we can see that the slope [tex]\( m_1 \)[/tex] of the given line is [tex]\( -\frac{1}{2} \)[/tex].
2. Determine the slope of the perpendicular line:
The slope of a line perpendicular to another line is the negative reciprocal of the slope of the given line. So, if the slope of the given line is [tex]\( m_1 = -\frac{1}{2} \)[/tex], then the slope [tex]\( m_2 \)[/tex] of the perpendicular line is:
[tex]\[ m_2 = -\frac{1}{m_1} = -\frac{1}{-\frac{1}{2}} = 2 \][/tex]
3. Use the point-slope form of the equation:
We have the slope [tex]\( m_2 = 2 \)[/tex] and the point [tex]\( (2, 7) \)[/tex]. The point-slope form of the equation of a line is given by:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
where [tex]\( (x_1, y_1) \)[/tex] is the point [tex]\((2, 7)\)[/tex].
4. Substitute the slope and point into the point-slope form:
[tex]\[ y - 7 = 2(x - 2) \][/tex]
5. Simplify the equation to slope-intercept form [tex]\( y = mx + b \)[/tex]:
[tex]\[ y - 7 = 2(x - 2) \][/tex]
[tex]\[ y - 7 = 2x - 4 \][/tex]
[tex]\[ y = 2x - 4 + 7 \][/tex]
[tex]\[ y = 2x + 3 \][/tex]
Therefore, the equation of the line perpendicular to [tex]\( y = -\frac{1}{2} x - 5 \)[/tex] and passing through the point [tex]\( (2, 7) \)[/tex] in slope-intercept form is:
[tex]\[ y = 2x + 3 \][/tex]
1. Identify the slope of the given line:
The equation of the given line is [tex]\( y = -\frac{1}{2} x - 5 \)[/tex]. From this equation, we can see that the slope [tex]\( m_1 \)[/tex] of the given line is [tex]\( -\frac{1}{2} \)[/tex].
2. Determine the slope of the perpendicular line:
The slope of a line perpendicular to another line is the negative reciprocal of the slope of the given line. So, if the slope of the given line is [tex]\( m_1 = -\frac{1}{2} \)[/tex], then the slope [tex]\( m_2 \)[/tex] of the perpendicular line is:
[tex]\[ m_2 = -\frac{1}{m_1} = -\frac{1}{-\frac{1}{2}} = 2 \][/tex]
3. Use the point-slope form of the equation:
We have the slope [tex]\( m_2 = 2 \)[/tex] and the point [tex]\( (2, 7) \)[/tex]. The point-slope form of the equation of a line is given by:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
where [tex]\( (x_1, y_1) \)[/tex] is the point [tex]\((2, 7)\)[/tex].
4. Substitute the slope and point into the point-slope form:
[tex]\[ y - 7 = 2(x - 2) \][/tex]
5. Simplify the equation to slope-intercept form [tex]\( y = mx + b \)[/tex]:
[tex]\[ y - 7 = 2(x - 2) \][/tex]
[tex]\[ y - 7 = 2x - 4 \][/tex]
[tex]\[ y = 2x - 4 + 7 \][/tex]
[tex]\[ y = 2x + 3 \][/tex]
Therefore, the equation of the line perpendicular to [tex]\( y = -\frac{1}{2} x - 5 \)[/tex] and passing through the point [tex]\( (2, 7) \)[/tex] in slope-intercept form is:
[tex]\[ y = 2x + 3 \][/tex]
Thanks for stopping by. We are committed to providing the best answers for all your questions. See you again soon. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Thank you for visiting Westonci.ca. Stay informed by coming back for more detailed answers.