Can you write an equation of a line that goes through the point 2 and negative 8 if the slope is undefined? No
It depends on what you mean by "undefined". If you mean that the slope is unknown then no, there are an infinite number of lines passing through a single point.
If you mean that the slope is of the form m = 1/0, then this is the case for a vertical line passing through the point (2,-8). The equation for this line is x = 2.
Let the required equation of a line be y =mx+c where m is any slope.
If this line passes through (x1,y1) then ,
(y- -8) = m(x-2) as (x1,y1)=(2,-8). Or
y=mx-2m+8 defines any straight line through (2,-8) for an desired slope.
Special cases when m is infinite:
If m is infinite, then 1/m = 0, then y/m=0 = x-2 or x-2 = 0 is a perpendicular line to x axis which passes through (2,-8)
Special case when m is zero:
If m= o, y=-8 is parallel to x axis.
Hope this helps.