Question

What is Joint Variation Theorem

Answer 1

Theorem of Joint Variation: If x ∝ y when z is constant and x ∝ z when y is constant, then x ∝ yz when both y and z vary.

Proof: 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. 

Therefore x ∝ yz when both y and z vary. 

Answer 2

If x ∝ y when z is constant and x ∝ z when y is constant, then x ∝ yz when both y and z vary.