It would seem that

Ax + By + C = 0

doesn’t really have a direction. Its just an infinite line, right?

WRONG.

take

-x + y = 0

This is the line through the origin, forming the line y = x.

1. Does this line have a direction?

2. Is there another line with a different equation that DESCRIBES THE SAME LINE?

Yes there is, to the second question, and YES IT DOES, to the first question.

The answer to 2.: The equivalent line is x – y = 0. That line is EXACTLY the same as -x + y = 0, only it “points” in the opposite direction.

So which “way” is -x + y pointing, and which way is x – y = 0 pointing?

Well, I’ll tell you.

x – y = 0 points “down and to the left”

-x + y = 0 points “up and to the right”

the reason for this is clear when you use vectors and the equations

A = y_{1} – y_{2}

B = x_{2} – x_{1}

C = x_{1}y_{2} – x_{2}y_{1}

Then, if you use the points

P_{1} = (1,1)

P_{2} = (0,0)

You get the equation

A = 1 – 0 = 1

B = 0 – 1 = -1

C = 0

Then

x – y = 0

POINTS DOWN AND LEFT!!

And so, the opposite equation

-x + y = 0

POINTS UP AND RIGHT!!

On the “left side” of a line is +

On the “right side” of a line is –

Taking the one that goes up and right:

plug in ( 2, 0 )

-x + y = 0

-2 + 0 = -2

So, (2,0) is negative in that line equation, meaning (2,0) is “right” of -x + y = 0

Taking (-2,0)

2 + 0 = 2

so, (-2,0) is “left” of -x + y = 0