Query regarding implicit differentiation

[t_n_p]

Homework Statement

Use implicit diff to find partial x and partial y if

x² + y² + z² = 3xyz

The Attempt at a Solution

firstly with partial x...

2x + 2z(∂z/∂x) = 3yz + 3xy(∂z/dx)
2x - 3yz = 3xy(∂z/dx) - 2z(∂z/∂x)
2x - 3yz = (3xy - 2z)(∂z/dx)
(2x - 3yz)/(3xy - 2z) = (∂z/dx)

yet in the solutions, they have moved all the (∂z/dx) terms to the left hand side thus giving

(∂z/dx) = (3yz - 2x)/(2z - 3xy)

Quite obviously this is not the same as my answer. My question is why should the two answers differ? MUST you take all (∂z/dx) terms to the left hand side as some sort of convention?

 

[tiny-tim]

(2x - 3yz)/(3xy - 2z) = (∂z/dx)

yet in the solutions, they have moved all the (∂z/dx) terms to the left hand side thus giving

(∂z/dx) = (3yz - 2x)/(2z - 3xy)

Quite obviously this is not the same as my answer.

Hi t_n_p!

They are the same!

Yours just has the top and bottom both multiplied by -1.

MUST you take all (∂z/dx) terms to the left hand side as some sort of convention?

erm … yes!

(unless you're writing in Hebrew or Arabic )

… you won't lose any marks for not doing it … but if you're asked an ordinary English question "what is an apple?" or "what is ∂z/∂x?", you start the answer with "an apple is …" or "∂z/∂x is …"

So ∂z/∂x goes on the left!

A math proof should sound like good English when you read it out!

 

[t_n_p]

hmmm, the thought never occurred to me...

also regarding the second point, I meant during the taking the dy/dx and non dy/dx terms to the left or rhs. obviously I would write the answer as dy/dx = ...

Just having one of those mindblocks...
Cheers