Respuesta :
Two perpendicular lines have reciprocal and opposite slopes.
First we have to write the given line in the slope-intercept form:
[tex]y=mx+b[/tex]Where m is the slope and b is the y-intercept.
We have this equation:
[tex]x-4y=20[/tex]To write it in the slope-intercept form we have to clear y:
[tex]\begin{gathered} x-20=4y \\ \downarrow \\ y=\frac{1}{4}x-5 \end{gathered}[/tex]The slope is 1/4 and the y-intercept is -5.
The slope of the perpendicular line will be the opposite and reciprocal of 1/4, that's -4.
For now we have the perpendicular line's equation:
[tex]y_p=-4x+b[/tex]There are a lot of lines that are perpendicular to the given line, but only one that passes through (2, -5). We use this point to find the y-intercept by replacing x = 2 and y = -5 into the expression above and solving for b:
[tex]\begin{gathered} -5=-4\cdot2+b \\ -5=-8+b \\ -5+8=b \\ b=3 \end{gathered}[/tex]The y-intercept of the perpendicular line is 3.
The equation of a line perpendicular to the given line that passes through the point (2,-5) is
[tex]y_p=-4x+3[/tex]