If x - y z and y - z x- then prove that z is a non-zero variation constant-

X ∝ y z ⇒ x=k_1· y z (k_1≠ 0=. variation constant ) ⇒ y=x/k_1 z.......(1) Again, y ∝ zx ⇒ y=k_2· z x (k_2≠ 0=. variation constant ) amp;⇒x/k_1 z=k_2· z x[ fro ...

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Standard VIII

Mathematics

Direct Variation

If x ∝ y z an...

Question

If $x \propto y z$ and $y \propto z x$ , then prove that $z$ is a non-zero variation constant.

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Solution

$x \propto y z \Rightarrow x = k_{1} \cdot y z (k_{1} \neq 0 =$ variation constant $)$
$\Rightarrow y = \frac{x}{k_{1} z} .......(1)$
Again, $y \propto z x \Rightarrow y = k_{2} \cdot z x (k_{2} \neq 0 =$ variation constant $)$
$\begin{matrix} \Rightarrow \frac{x}{k_{1} z} = k_{2} \cdot z x [from (1)] \Rightarrow z^{2} = k_{1} k_{2} \Rightarrow z = \sqrt{k_{1} k_{2}} .......(2) \end{matrix}$
Since $k_{1}, k_{2}$ are non-zero variation constants,
$∴ \sqrt{k_{1} k_{2}}$ is a non-zero variation constant.
$\Rightarrow z$ is a ron-zero variation constant.

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If x∝yz and y∝zx, then prove that z is a non-zero variation constant.

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