Let and .
Let two lines given by
y1 = ax+b
y2 = cx+d
they are perpendicular, if a*c = -1
they are parallel, if a=c (and b is not equal to d)
Now back to your excersice
(1)y =2/3 x + 3 and y = -3x + 2
It is a = 2/3 and c = -3. It is obvious that these lines are not parallel
But it is a*c = 2/3 * (-3) = -2
This is not equal to -1, because
So they are not perp.
2x + 5y = 1 and y = 5/2x +4
Solve the first "line" for y
2x + 5y = 1
5y = 1-2x
y =1/5 * (1-2x) = 1/5 - 2/5x = -2/5x+1/5
Okay, and the other line is defined by y = 5/2x +4
It is a = -2/5 and c = +5/2 => they are not parallel, but perpendicular.
They are the same, if
a) the lines are parallel, that means a=c, e. g.
b) b = d, e. g. in red_dog 's post the criteria is
'Neither' is the right answer for (1)
I'm glad you didn't go offline and checked the forum again.