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Since x ∝ y when z is constant Therefore x = ky where k = constant of variation and is independent to the changes of x and y.

Again, x ∝ z when y is constant.

or, ky ∝ z when y is constant (since, x = ky).

or, k ∝ z (y is constant).

or, k = mz where m is a constant which is independent to the changes of k and z.

Now, the value of k is independent to the changes of x and y. Hence, the value of m is independent to the changes of x, y and z.

Therefore x = ky = myz (since, k = mz)

where m is a constant whose value does not depend on x, y and z.

Certified answers contain reliable, trustworthy information vouched for by a hand-picked team of experts. Brainly has millions of high quality answers, all of them carefully moderated by our most trusted community members, but certified answers are the finest of the finest.

Given that,

x ∝ y when z is constant,

∴ x = ky (where k = constant and it is independent of x and y.)

Also, x ∝ z provided that y is constant.

⇒ ky ∝ z when y is constant (∵ x = ky).

⇒ k ∝ z (y is constant).

⇔ k = mz (where m is a constant and it is independent of k and z.) ⇒ The value of k is independent of x and y. ∴ The value of m is independent to the changes of x, y and z.

∴ x = ky = mzy (since, k = mz)

(where m is a constant and its value does not depend on x, y and z. )