Question

Write the equation of a line perpendicular to 9xplus7yequals6 that passes through the point ​(9​,negative 7​).

Answer 1

$k:\ y=m_1x+b_1\\\\l:\ y=m_2x+b_2\\\\k\ ||\ l\iff m_1=m_2\\\\k\ \perp\ l\iff m_1m_2=-1$

We have:

$k:\ 9x+7y=6\ \ \ |-9x\\\\7y=-9x+6\ \ \ \ |:7\\\\y=-\dfrac{9}{7}x+\dfrac{6}{9}\to m_1=-\dfrac{9}{7}\\\\l:\ y=m_2x+b\\\\l\ \perp\ k\iff-\dfrac{9}{7}m_2=-1\ \ \ \ |\cdot\left(-\dfrac{7}{9}\right)\\\\m_2=\dfrac{7}{9}\to\ l:\ y=\dfrac{7}{9}x+b$

We know. The line l passes throught the point (9, -7). Substitute the coordinates of the poin to the equation of line l :

$-7=\dfrac{7}{9}\cdot9+b\\\\-7=7+b\ \ \ \ |-7\\\\b=-14$

$y=\dfrac{7}{9}x-14\ \ \ \ |\cdot9\\\\9y=7x-126\ \ \ \ |-9y\ |+126\\\\7x-9y=126$

Answer: $y=\dfrac{7}{9}x-14\to7x-9y=126$