How would i know what lines are intersecting, perpendicular, parallel and coinciding based off an equation given? like in the picture listed.

If lines are paralel, then they have the same slope
if the lines are pependicuilar, the slopes mutiply to -1

so

y=mx+b
m=slope

and
ax+by=c
-a/b is the slope

given
y=2/3x-2
slope is 2/3

x+y=4
slope is -1/1 or -1

2/3≠-1 and -1 times 2/3≠-1 so they are neither perpendicular or paralell

since they aren't paralell, we know they intersect somewhere