Can Proportional Variation be Deduced Algebraically from Given Equations?

[dx]

Consider the following situation :

There are three variables A, B and C. (and i,j,k are constants )
Keeping C constant, and varying the other two, you find that

A = kB ------(1)

Now, Keeping B constant, and varying the other two, you find that

A = iC ------(2)

I know that it follows from these two observations that

A = jBC

But I am not sure how we can algebraically deduce this from the equations (1) and (2).

We get $$A^2 = ikBC$$

Presumably, $$ik = \sqrt{j}BC$$
But how do we deduce this?

 

[Tide]

You have

$$A = k(C) \times B = i(B) \times C$$

where k and i are functions of C and B, respectively. If B and C are independent variables then the only way k(C)B and i(B)C can be equal is if k is proportional to C and i is proportional to B. Therefore, A = jBC.

 

[dx]

Thanks for the help.