Sometimes in algebraic exercises where the solution is a ratio I get the solution multiplied by $(-1/-1)$.
For example I get $$ (2zy-y) / (3-z) = x $$ If I multiply the fraction by $(-1/-1)$ I get $$ (y-2zy) / (z-3) = x $$ This can be quite confusing If I have large algebraic terms. Is there a rule to follow which determines how the solution should be displayed.
I realize that this is a really low level question but I could not find any guidelines how to handle this...
For some things there is a "canonical form", and perhaps the principal utility of canonical forms is that you can tell whether two things are equal by seeing whether they're the same once they're in the canonical form. (Simplest radical form is among the most familiar examples of this; putting fractions in lowest terms is another.)
For expressions like your examples, I know of no canonical form. Seeing whether your answer is the same as someone else's (e.g. one in the back of the book) may take some work sometimes. Sometimes on this site someone wonders about things like why they keep getting $\tan^2 x$ as an antiderivative when the answer in the back of the book is $\sec^2 x$; the answer is that because of trigonometric identities $\tan^2 x + \text{constant}$ is the same as $\sec^2 x+\text{constant}$ (but the two "constants" differ from each other by $1$).
Sometimes there are reasons to prefer one form. It can be something as dumb-but-valid as preferring $\dfrac 1 {b-a}$ to $\dfrac 1 {-a+b}$ because it's typographically simpler and better-looking. Another reason is stylistic consistency; for example this form of this identity $$ \left( \frac{\sqrt 2\,\cos x -1 }{\cos x-\sqrt 2} \right)^2 + \left( \frac{\sin x}{\cos x - \sqrt 2}\right)^2 = 1 $$ is better than the form $$ \left( \frac{\sqrt 2\,\cos x -1 }{\cos x-\sqrt 2} \right)^2 + \left( \frac{\sin x}{\sqrt 2 - \cos x}\right)^2 = 1 $$ because the denominators look the same. If the reader must stop for a fraction of a second to realize these both say the same thing, then the reader is distracted from the brilliant rhythmic chain of thought that your writing exhibits.
How should one write Heron's formula? Should one write $$ \text{area} = \frac 1 4 \sqrt{\vphantom{\frac 1 1} (a+b+c)(-a+b+c)(a-b+c)(a+b-c) } \text{ ?} $$ Or is it better to write $$ \text{area} = \frac 1 4 \sqrt{\vphantom{\frac 1 1} (a+b+c)(a+b-c)(b+c-a)(c+a-b) } \text{ ?} $$ I would claim that that last is superior to this: $$ \text{area} = \frac 1 4 \sqrt{\vphantom{\frac 1 1} (a+b+c)(a+b-c)(b+c-a)(a+c-b) } \text{ .} $$ See if you can figure out why.
