# Are $x=-\frac{m}{n}$ and $-x=\frac{m}{n}$ the same?

I was wondering that is

$x=-\frac{m}{n}$

same as

$-x=\frac{m}{n}$

The question popped into my mind when had

$x=-\frac{11}{14}$ or $-x=\frac{11}{14}$

as an anwser to one of my equations. Was the $x$ positive or negative $-x$ only depended on which side I putted numbers and X's in my calculations.

• When solving an equation, you're usually expected to denote the value of $x$, not the value of $-x$. – barak manos Jan 14 '16 at 10:22
• If you want to quote the value of $x$ then you would write $x = -11/15$ (which is negative). By writing $-x = 11/15$ then you are saying that (the different number) $-x$ equals to $11/15$ (which is positive). – Winther Jan 14 '16 at 10:27
• Try Googling "multiplicative reflexive axiom". Reason as follows $$x = - \frac{m}{n}\implies cx= c(-\frac{m}{n})$$ Now let $c = -1$ – John Joy Jan 14 '16 at 14:32

## 2 Answers

In both cases the sign of $x$ is the same. If $\frac{m}{n}>0$ Then $$x = -\frac{m}{n}<0 \Rightarrow x<0$$ In the second equation $-x = \frac{m}{n}$ we also have that $x<0$ since $$-x = \frac{m}{n} \Rightarrow x = -\frac{m}{n}$$ The same can be done when $\frac{m}{n}<0$.

They are the same in the sense that those equations are equivalent, i.e. they have the same solutions.