Welcome to Westonci.ca, your ultimate destination for finding answers to a wide range of questions from experts. Ask your questions and receive accurate answers from professionals with extensive experience in various fields on our platform. Get quick and reliable solutions to your questions from a community of experienced experts on our platform.
Sagot :
To find the equation of a line that is perpendicular to the given line [tex]\(2x + y = -5\)[/tex] and passes through the point [tex]\((-1, -2)\)[/tex], follow these steps:
1. Determine the slope of the given line [tex]\(2x + y = -5\)[/tex]:
- Rewrite the equation in slope-intercept form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope.
- [tex]\(2x + y = -5\)[/tex] can be rewritten as [tex]\(y = -2x - 5\)[/tex].
- Therefore, the slope [tex]\(m\)[/tex] of the given line is [tex]\(-2\)[/tex].
2. Find the slope of the line perpendicular to the given line:
- The slope of a line perpendicular to another is the negative reciprocal of the slope of the original line.
- The negative reciprocal of [tex]\(-2\)[/tex] is [tex]\(\frac{1}{2}\)[/tex].
3. Form the equation of the line in slope-intercept form [tex]\(y = mx + b\)[/tex]:
- Use the point-slope form of the line equation:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
- Substitute the slope ([tex]\(\frac{1}{2}\)[/tex]) and the given point [tex]\((-1, -2)\)[/tex]:
[tex]\[ y - (-2) = \frac{1}{2}(x - (-1)) \][/tex]
[tex]\[ y + 2 = \frac{1}{2}(x + 1) \][/tex]
4. Solve for the [tex]\(y\)[/tex]-intercept [tex]\(b\)[/tex]:
- Distribute and solve for [tex]\(y\)[/tex]:
[tex]\[ y + 2 = \frac{1}{2}x + \frac{1}{2} \][/tex]
[tex]\[ y = \frac{1}{2}x + \frac{1}{2} - 2 \][/tex]
[tex]\[ y = \frac{1}{2}x - \frac{3}{2} \][/tex]
Therefore, the equation of the line perpendicular to [tex]\(2x + y = -5\)[/tex] that passes through the point [tex]\((-1, -2)\)[/tex] is [tex]\(y = \frac{1}{2}x - \frac{3}{2}\)[/tex].
Among the given options, the correct equation is:
[tex]\[ \boxed{y = \frac{1}{2}x - \frac{3}{2}} \][/tex]
1. Determine the slope of the given line [tex]\(2x + y = -5\)[/tex]:
- Rewrite the equation in slope-intercept form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope.
- [tex]\(2x + y = -5\)[/tex] can be rewritten as [tex]\(y = -2x - 5\)[/tex].
- Therefore, the slope [tex]\(m\)[/tex] of the given line is [tex]\(-2\)[/tex].
2. Find the slope of the line perpendicular to the given line:
- The slope of a line perpendicular to another is the negative reciprocal of the slope of the original line.
- The negative reciprocal of [tex]\(-2\)[/tex] is [tex]\(\frac{1}{2}\)[/tex].
3. Form the equation of the line in slope-intercept form [tex]\(y = mx + b\)[/tex]:
- Use the point-slope form of the line equation:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
- Substitute the slope ([tex]\(\frac{1}{2}\)[/tex]) and the given point [tex]\((-1, -2)\)[/tex]:
[tex]\[ y - (-2) = \frac{1}{2}(x - (-1)) \][/tex]
[tex]\[ y + 2 = \frac{1}{2}(x + 1) \][/tex]
4. Solve for the [tex]\(y\)[/tex]-intercept [tex]\(b\)[/tex]:
- Distribute and solve for [tex]\(y\)[/tex]:
[tex]\[ y + 2 = \frac{1}{2}x + \frac{1}{2} \][/tex]
[tex]\[ y = \frac{1}{2}x + \frac{1}{2} - 2 \][/tex]
[tex]\[ y = \frac{1}{2}x - \frac{3}{2} \][/tex]
Therefore, the equation of the line perpendicular to [tex]\(2x + y = -5\)[/tex] that passes through the point [tex]\((-1, -2)\)[/tex] is [tex]\(y = \frac{1}{2}x - \frac{3}{2}\)[/tex].
Among the given options, the correct equation is:
[tex]\[ \boxed{y = \frac{1}{2}x - \frac{3}{2}} \][/tex]
Thanks for using our service. We aim to provide the most accurate answers for all your queries. Visit us again for more insights. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. We're dedicated to helping you find the answers you need at Westonci.ca. Don't hesitate to return for more.