Question

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

Answer 1

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

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. )

∴ x ∝ yz when both y and z vary.