Direct Proportions and Their ConstantsDate: 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 and x = z*C then 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 http://mathforum.org/dr.math/ 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! |
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