Math: Scalar Equation of a Plane

Determine whether the following pairs of planes are coincident, parallel and distinct or neither.

a) x + 3y - z - 2 = 0 and 2x + 6y - 2z - 8 = 0

TEXTBOOK ANSWER: parallel and distinct

Please provide a step by step solution so I can understand the question...

Thanks.

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  1. very simple

    notice that if you divide the second equation by 2 you get
    x + 3y - z - 4 = 0

    compare this with the first to see that they have the same normal, so they are parallel.
    Since their constants are differenct they must form two parallel planes.
    I call that the "floor-ceiling" situation.

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