Find the equation for the line that passes through the point (1, −4) , and that is perpendicular to the line with the equation x = 4

We are asked to find the equation of the line through the point (1,-4) and perpendicular to the line x=4:

To write an equation of a line we need either a point and the slope, or two points. Here we are given a point so it suffices to find the slope.

Since the line x=4 has no slope (the slope is undefined for a vertical line), the line perpendicular will have slope m=0 (a horizontal line.)

Thus we have a line with slope 0 going through the point (1,-4).

The equation is y=-4. (Or y=0(x)-4, but 0*x=0 for all x, so we leave that term out.)