Respuesta :
Answer:
[tex]y = -\frac{3}{2}x+6[/tex]
Step-by-step explanation:
Given:
The given equation of the line [tex]y = \frac{2}{3} x + 5[/tex] that passes through the point (4, 0).
Part A.
The equation of the line.
[tex]y=mx+c[/tex]-----------(1)
Where:
m = Slope of the line
c = y-intercept
The given equation of the line.
[tex]y = \frac{2}{3} x + 5[/tex]
Comparing the given equation with equation 1.
The slope of the line is [tex]m=\frac{2}{3}[/tex] and y-intercept [tex]c = 5[/tex]
We know that the slope of the perpendicular line is [tex](-\frac{1}{m})[/tex].
So the slope of the perpendicular line is [tex](m=-\frac{1}{\frac{2}{3}}=-\frac{3}{2})[/tex].
Using point slope formula we write the equation of the perpendicular line that passes through the point (4, 0).
[tex]y-y_{1} = m(x-x_{1}[/tex]
Now we substitute the slope of the perpendicular line [tex]m=-\frac{3}{2}[/tex] and [tex]y_{1} = 0, x_{1}=4[/tex] from point (4, 0) in above equation.
[tex]y-0 = -\frac{3}{2}(x-4)[/tex]
[tex]y = -\frac{3}{2}x-(-\frac{3}{2}\times 4)[/tex]
[tex]y = -\frac{3}{2}x-(-3\times 2)[/tex]
[tex]y = -\frac{3}{2}x-(-6)[/tex]
[tex]y = -\frac{3}{2}x+6[/tex]
Therefore the perpendicular line equation is [tex]y = -\frac{3}{2}x+6[/tex].
Part B.
1. The slope of the perpendicular line is [tex]m=-\frac{3}{2}[/tex].
2. The y-intercept of the perpendicular line is 6.
3. The equation of the line [tex]y = -\frac{3}{2}x+6[/tex] is perpendicular to the equation of line [tex]y = \frac{2}{3} x + 5[/tex].
The equation of line is [tex]y=-\frac{3}{2}x+6[/tex] , which is perpendicular to the line [tex]y=\frac{2}{3}x+5[/tex]
(A). Equation of given line is, [tex]y=\frac{2}{3}x+5[/tex]
Comparing above equation of line from slope - intercept equation of line [tex]y=mx+c[/tex]
So, slope of given line is, [tex]m=\frac{2}{3}[/tex]
Since, The perpendicular line slope will be the opposite reciprocal of the original slope.
So, slope of perpendicular line is, = [tex]-\frac{1}{m}[/tex]
Slope = [tex]-\frac{1}{\frac{2}{3} }=-\frac{3}{2}[/tex]
Equation of line perpendicular to given line, [tex]y=-\frac{3}{2} x+c[/tex]
Since, line passing through (4, 0)
Thus, [tex]0=-\frac{3}{2}(4)+c\\\\c=6[/tex]
Therefore, equation of line perpendicular to given line is, [tex]y=-\frac{3}{2}x+6[/tex]
(B).The perpendicular line slope will be the opposite reciprocal of the original slope.
Equation of perpendicular line in slope intercept form,, [tex]y=-\frac{3}{2}x+6[/tex]
Y - intercept is 6.
Slope of perpendicular line is [tex]-\frac{3}{2}[/tex]
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