Question

Write the equation of a line that is perpendicular to the given line and that passes through the given point. -x+5y=14; (-5,-2)

Answer 1

$Let\\\\k:y=m_1x+b_1\\\\l:y=m_2x+b_2\\\\l\ \perp\ k\iff m_1m_2=-1\\\\\text{We have}\ k:-x+5y=14\qquad|+x\\\\5y=x+14\qquad|:5\\\\y=\dfrac{1}{5}x+\dfrac{14}{5}\to m_1=\dfrac{1}{5}\\\\l:y=mx+b\to\dfrac{1}{5}m=-1\qquad|\cdot5\\\\m=-5\\\\l:y=-5x+b\\\\\text{Put the coordinates of the point (-5, -2) to the equation of the line}\ l:\\\\-2=-5(-5)+b\\-2=25+b\qquad|-25\\b=-27\\\\Answer:\ y=-5x-27\to5x+y=-27$