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Direct Proportions and Their Constants

Date: 01/12/2014 at 05:35:34
From: Akash
Subject: If x=y*K and x=z*C, how does x=P*y*z(capitals are constants)

In science, one often sees that if one quantity is directly proportional
to two or more other quantities independently, it must be proportional to
the product of all the quantities.

In other words, for constants K, C, and P, if 

   x = y*K


   x = z*C


   x = P*y*z

How does one justify this mathematically?

Since x = K*y and x = C*z, doesn't that imply that x^2 = P*y*z (by
multiplying the two equations)? How do you get x = P*y*z?

Intuitively, the law must hold, but I can't think of a way of justifying
it mathematically.

Date: 01/12/2014 at 21:09:13
From: Doctor Peterson
Subject: Re: If x=y*K and x=z*C, how does x=P*y*z(capitals are constants)

Hi, Akash.

Suppose that x = Ay and x = Bz, where A is constant WHEN z IS FIXED, and B
is constant WHEN y IS FIXED. That's what you mean by "constant" here. If A
were absolutely constant, then x would not vary with z; and similarly for
B. This is why you can't multiply the equations as you did; since A and B
actually depend on z and y respectively, P = AB would not really be a
constant at all.

So we might write the relations this way:

   x = A(z) y
   x = B(y) z

Here, A(z) and B(y) are functions that depend only on the indicated
variables. Now, set the two expressions for x equal:

   A(z) y = B(y) z

For fixed y, the right side is proportional to z; therefore the left side
must be as well. So A(z) must be some constant C times z; similarly, B(y)
must be some constant D times y:

   A(z) = Cz
   B(y) = Dy

But then

   x = A(z) y = Cz * y = Cyz

   x = B(y) z = Dy * z = Dyz

So we see that C and D are actually the same constant, and x has the form

   x = Pyz

Going in the other direction, we can more easily see that IF x = Pyz, then

   x = (Py)z

Consequently, x is directly proportional to z when y stays the same. 

Similarly, since ...

   x = (Pz)y ,

... x is directly proportional to y when z is held constant.

So x is directly proportional to both y and z if and only if it is
directly proportional to yz.

This is also called joint variation: We say that x varies jointly as y and
z when it varies directly as each of them individually, and in that case
we can write x = kyz.

- Doctor Peterson, The Math Forum 

Date: 02/14/2014 at 08:04:30
From: Akash
Subject: Thank you (If x=y*K and x=z*C, how does x=P*y*z(capitals are constants))

Thanks a lot, Dr. Peterson!
Associated Topics:
High School Linear Equations

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