Discover new knowledge and insights with IDNLearn.com's extensive Q&A database. Get prompt and accurate answers to your questions from our community of experts who are always ready to help.
Sagot :
To find the equation of the line that is perpendicular to the given line [tex]\(2x + 12y = -1\)[/tex] and passes through the point [tex]\((0, 9)\)[/tex], we need to follow these steps:
1. Convert the given line equation to slope-intercept form ([tex]\(y = mx + b\)[/tex]):
[tex]\[ 2x + 12y = -1 \][/tex]
Solve for [tex]\(y\)[/tex]:
[tex]\[ 12y = -2x - 1 \][/tex]
[tex]\[ y = \frac{-2}{12}x - \frac{1}{12} \][/tex]
Simplify the coefficients:
[tex]\[ y = -\frac{1}{6}x - \frac{1}{12} \][/tex]
So the slope [tex]\(m\)[/tex] of the given line is [tex]\(-\frac{1}{6}\)[/tex].
2. Find the slope of the line that is perpendicular to the given line:
The slope of a line perpendicular to another is the negative reciprocal of the slope of the original line. Therefore, the perpendicular slope [tex]\(m_{\perp}\)[/tex] is:
[tex]\[ m_{\perp} = -\frac{1}{-\frac{1}{6}} = 6 \][/tex]
3. Use the slope-intercept form of the line equation ( [tex]\(y = mx + b\)[/tex] ) and the given point [tex]\((0, 9)\)[/tex] to find the y-intercept [tex]\(b\)[/tex]:
Substitute the point [tex]\((0, 9)\)[/tex] into the equation with the perpendicular slope:
[tex]\[ y = 6x + b \][/tex]
Using the point [tex]\((0, 9)\)[/tex]:
[tex]\[ 9 = 6 \cdot 0 + b \][/tex]
[tex]\[ b = 9 \][/tex]
4. Write the final equation of the line:
The equation of the line that is perpendicular to the given line and passes through the point [tex]\((0, 9)\)[/tex] is:
[tex]\[ y = 6x + 9 \][/tex]
Thus, the equation in slope-intercept form of the line that is perpendicular to the given line and passes through the point [tex]\((0,9)\)[/tex] is:
[tex]\[ \boxed{y = 6x + 9} \][/tex]
1. Convert the given line equation to slope-intercept form ([tex]\(y = mx + b\)[/tex]):
[tex]\[ 2x + 12y = -1 \][/tex]
Solve for [tex]\(y\)[/tex]:
[tex]\[ 12y = -2x - 1 \][/tex]
[tex]\[ y = \frac{-2}{12}x - \frac{1}{12} \][/tex]
Simplify the coefficients:
[tex]\[ y = -\frac{1}{6}x - \frac{1}{12} \][/tex]
So the slope [tex]\(m\)[/tex] of the given line is [tex]\(-\frac{1}{6}\)[/tex].
2. Find the slope of the line that is perpendicular to the given line:
The slope of a line perpendicular to another is the negative reciprocal of the slope of the original line. Therefore, the perpendicular slope [tex]\(m_{\perp}\)[/tex] is:
[tex]\[ m_{\perp} = -\frac{1}{-\frac{1}{6}} = 6 \][/tex]
3. Use the slope-intercept form of the line equation ( [tex]\(y = mx + b\)[/tex] ) and the given point [tex]\((0, 9)\)[/tex] to find the y-intercept [tex]\(b\)[/tex]:
Substitute the point [tex]\((0, 9)\)[/tex] into the equation with the perpendicular slope:
[tex]\[ y = 6x + b \][/tex]
Using the point [tex]\((0, 9)\)[/tex]:
[tex]\[ 9 = 6 \cdot 0 + b \][/tex]
[tex]\[ b = 9 \][/tex]
4. Write the final equation of the line:
The equation of the line that is perpendicular to the given line and passes through the point [tex]\((0, 9)\)[/tex] is:
[tex]\[ y = 6x + 9 \][/tex]
Thus, the equation in slope-intercept form of the line that is perpendicular to the given line and passes through the point [tex]\((0,9)\)[/tex] is:
[tex]\[ \boxed{y = 6x + 9} \][/tex]
We appreciate your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. IDNLearn.com is dedicated to providing accurate answers. Thank you for visiting, and see you next time for more solutions.