Calculus 10th Edition

Published by Brooks Cole
ISBN 10: 1-28505-709-0
ISBN 13: 978-1-28505-709-5

Chapter 13 - Functions of Several Variables - 13.1 Exercises - Page 876: 3


$z$ is a function of $x$ and $y$.

Work Step by Step

For $z$ to be the function of $x$ and $y$ then for each $x$ and $y$ we need to have only one value of $z$. Here we have $$x^2 z+3y^2-xy=10.$$ Solving for $z$: $$x^2 z+3y^2-xy=10\Rightarrow x^2z=10+xy-3y^2\Rightarrow z=\frac{10+xy-3y^2}{x^2}$$ The last equality says that for each $x$ and $y$ we have ONLY ONE value for $z$ which is calculated from that equality so $z$ is a function of $x$ and $y$.
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