How to go from the explicit form of a straight line to the implicit form?

You can consider the explicit form as two equations:

1. x = a + r
2. y = b + rd

– If c=d=0, you do not have a straight line, but only a point (a, b).

– If c=0 (and d≠0), equation 1. The other equation then only says “no one r, then you will get any y (or x)”.

– For c≠0 and d≠0, you get the implicit equation via d·(1.) − c·(2.):

• dx − cy = da − cb

(drc-crd rises).

For your example, 4×-3y=4.1-3.2=-2 results.

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If you look more closely at the implicit equation, you will see the vector (d, −c) perpendicular to (c, d) and the starting point (a, b). And the implicit equation simply says:

• (d, −c)(x, y) = (d, −c)(a, b).

or in vector writing short and crisp:

• n.e.c.