Question

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

Answer 1

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