Two perpendicular lines intersect on the y -axis. the equation of one line is x + 4y - 24 = 0 . determine the equation of the other line.

Intersecting on the y axis means x=0.
Plug x=0 in the equation to find out the y value of the intercept> 0+4y-24=0, y=6
so the both line passes the point (0,6)

two perpendicular lines have slopes that are negative reciprocals of each other, that is, if one slope is a/b, the other is -b/a

In this case, the line x+4y-24=0, 4y=-x+24, y=-1/4 x + 6, the slope is -1/4, so the slope of the perpendicular line is 4/1.

the equation is y=4x+6