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=-(9)/(7)x+(6)/(9)\to m_1=-(9)/(7)\\\\l:\ y=m_2x+b\\\\l\ \perp\ k\iff-(9)/(7)m_2=-1\ \ \ \ |\cdot\left(-(7)/(9)\right)\\\\m_2=(7)/(9)\to\ l:\ y=(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=(7)/(9)\cdot9+b\\\\-7=7+b\ \ \ \ |-7\\\\b=-14`

`y=(7)/(9)x-14\ \ \ \ |\cdot9\\\\9y=7x-126\ \ \ \ |-9y\ |+126\\\\7x-9y=126`

Answer:
`y=(7)/(9)x-14\to7x-9y=126`