First things first, let's get y as the subject of the equation
[tex]3x-4y=12 \\ 4y=3x-12 \\ y= \frac{3}{4}x-3 [/tex]
The perpendicular to a line involves flipping the fraction and changing the sign, i.e. 3/4 becomes -4/3
This however just gives us the gradient of the perpendicular, so now we must sub in the coordinates to work out the y intercept (i.e. c in y=mx+c)
[tex]7= \frac{-4}{3}(12)+c \\ 7=-16+c \\ c=23 [/tex]
Now we know in the case of the perpendicular passing through those exact coordinates, it intercepts the y axis at 23. Putting this all together:
[tex]y= \frac{-4}{3}x+23 [/tex]
