Family of lines equation, need help

Question

I'm super confused with this question: a family of lines have equations of the form 3x+by=12, where b is a non-zero number.

Find the equation of the line perpendicular to 3x+by=12 at the point (4,0).

I got the answer by=(1/3)x+12 but it says it's y=bx/3-4b/3.

Please help, thanks!

Answer 1

Convert Standard Form into slope-intercept form:

3x+by=12

y = (-3/b)x +12/b

Slope of parallel lines is negative reciprocal of this slope (-3/b):

y = (b/3)x + C [note: y-intercept is different,;not 12 ]

and we need point (4,9) to find C [note: we must use y=mx+C because b is already used.]

0 = (b/3)(4) + C

-4b/3 = C

so, the parallel line is:

y = (b/3)x - 4b/3 [ also y = bx/3 - 4b/3 ]