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.
