Join the conversation on IDNLearn.com and get the answers you seek from experts. Ask your questions and receive comprehensive, trustworthy responses from our dedicated team of experts.
Sagot :
To find the equation of the line that passes through the point [tex]\( (2,0) \)[/tex] and is perpendicular to the line [tex]\( y = -\frac{2}{5}x \)[/tex], we can follow these steps:
1. Identify the slope of the given line:
The given line is [tex]\( y = -\frac{2}{5}x \)[/tex]. The slope (m) of this line is [tex]\( -\frac{2}{5} \)[/tex].
2. Find the slope of the perpendicular line:
The slope of a line perpendicular to another is the negative reciprocal of the original slope. The negative reciprocal of [tex]\( -\frac{2}{5} \)[/tex] is [tex]\( \frac{5}{2} \)[/tex].
Thus, the slope of the perpendicular line is [tex]\( \frac{5}{2} \)[/tex].
3. Use the point-slope form to find the equation of the perpendicular line:
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 a point on the line and [tex]\( m \)[/tex] is the slope.
Given the point [tex]\( (2,0) \)[/tex] and the slope [tex]\( \frac{5}{2} \)[/tex], we can substitute these values into the point-slope form:
[tex]\[ y - 0 = \frac{5}{2} (x - 2) \][/tex]
4. Simplify the equation:
Simplifying the right side of the equation:
[tex]\[ y = \frac{5}{2} x - \frac{5}{2} \cdot 2 \][/tex]
[tex]\[ y = \frac{5}{2} x - 5 \][/tex]
Therefore, the equation of the line that passes through the point [tex]\( (2,0) \)[/tex] and is perpendicular to the line [tex]\( y = -\frac{2}{5}x \)[/tex] is:
[tex]\[ y = \frac{5}{2} x - 5 \][/tex]
5. Result:
The slope of the perpendicular line is [tex]\(\frac{5}{2}\)[/tex] and the y-intercept (where the line crosses the y-axis) is [tex]\(-5\)[/tex]. Thus, the equation of the perpendicular line in slope-intercept form is:
[tex]\[ y = \frac{5}{2} x - 5 \][/tex]
Putting these together, the perpendicular line has a slope of 2.5 and a y-intercept of -5.0.
1. Identify the slope of the given line:
The given line is [tex]\( y = -\frac{2}{5}x \)[/tex]. The slope (m) of this line is [tex]\( -\frac{2}{5} \)[/tex].
2. Find the slope of the perpendicular line:
The slope of a line perpendicular to another is the negative reciprocal of the original slope. The negative reciprocal of [tex]\( -\frac{2}{5} \)[/tex] is [tex]\( \frac{5}{2} \)[/tex].
Thus, the slope of the perpendicular line is [tex]\( \frac{5}{2} \)[/tex].
3. Use the point-slope form to find the equation of the perpendicular line:
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 a point on the line and [tex]\( m \)[/tex] is the slope.
Given the point [tex]\( (2,0) \)[/tex] and the slope [tex]\( \frac{5}{2} \)[/tex], we can substitute these values into the point-slope form:
[tex]\[ y - 0 = \frac{5}{2} (x - 2) \][/tex]
4. Simplify the equation:
Simplifying the right side of the equation:
[tex]\[ y = \frac{5}{2} x - \frac{5}{2} \cdot 2 \][/tex]
[tex]\[ y = \frac{5}{2} x - 5 \][/tex]
Therefore, the equation of the line that passes through the point [tex]\( (2,0) \)[/tex] and is perpendicular to the line [tex]\( y = -\frac{2}{5}x \)[/tex] is:
[tex]\[ y = \frac{5}{2} x - 5 \][/tex]
5. Result:
The slope of the perpendicular line is [tex]\(\frac{5}{2}\)[/tex] and the y-intercept (where the line crosses the y-axis) is [tex]\(-5\)[/tex]. Thus, the equation of the perpendicular line in slope-intercept form is:
[tex]\[ y = \frac{5}{2} x - 5 \][/tex]
Putting these together, the perpendicular line has a slope of 2.5 and a y-intercept of -5.0.
We value your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Your questions find clarity at IDNLearn.com. Thanks for stopping by, and come back for more dependable solutions.
