Explanation Proof: The function is, a x + b y + c z = d . The level curves of the function a x + b y + c z = d are described by the equation as follows. a x + b y + c z 0 = d a x + b y = d − c z 0 b y = − a x + d − c z 0 y = − a b x + d b − c b z 0 The equation is y = − a b x + d b − c b z 0 where z 0 is a constant in the range of f ( x , y ) . For z 0 , the level curves are line level curves in x y -plane . Let z 0 = 1 , then the level curve is aline and the slope m = − a b

Calculus: Early Transcendentals (2nd Edition)

2nd Edition

ISBN: 9780321947345

Author: William L. Briggs, Lyle Cochran, Bernard Gillett

Publisher: PEARSON

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Chapter 12.2, Problem 65E

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To prove: The level curvesof the plane ax+by+cz=d are parallel lines in the xy-plane provided a2+b2≠0 and c≠0 .

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Chapter 12.2, Problem 64E

Chapter 12.2, Problem 66E

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To prove: The level curvesof the plane a x + b y + c z = d are parallel lines in the x y -plane provided a 2 + b 2 ≠ 0 and c ≠ 0 .

Solution Summary: The author explains how the level curves of the function ax+by+cz=d are described by the equation as follows.