Why do I have to multiply by -1 when simplifying (sometimes)

[Jherek]

Homework Statement

Solve for s:
$$-4su+7t+9u+2=6t+4u-9$$

Homework Equations

The Attempt at a Solution

My answer:
$$s = \frac{-t-5u-11}{-4u}$$
The given answer:
$$s=\frac{t+5u+11}{4u}$$
I know that the right hand side has been multiplied by -1, but this raises some questions in my mind; Why do I have to do this, is my answer wrong and why isn't the left hand side also multiplied by -1?
Thanks for your help...

 

[Muphrid]

They haven't multiplied anything by -1. They factored -1 out from both the numerator and denominator and canceled it. Your answer is equivalent, but it's pretty typical to not leave any minus signs in denominators when simplifying.

 

[Jherek]

Ah. So that explains why the left-hand side is left untouched. It looks like the right hand side has been multiplied by -1 and that is how I've heard it explained before - but what you say makes more sense!
So I wasn't actually wrong as such, just not following convention.
Thank you.

 

[NascentOxygen]

Ah. So that explains why the left-hand side is left untouched. It looks like the right hand side has been multiplied by -1 and that is how I've heard it explained before

I suspect that what you were told is that on the right-hand side "we can multiply both "top" and "bottom" by -1[/color]". This is equivalent, if you think about it, to multiplying by 1 since (-1)/(-1) = 1. And when anything is multiplied by 1 you don't change its value. Hence, the left-hand side remains unchanged when you multiply the right-hand side by 1. (Remember, -1/-1 = 1)

 

[SammyS]

After all, which is simpler, $\displaystyle \frac{\,-3}{\,-4}$ or $\displaystyle \frac{3}{4}\ ?$

 

[Jherek]

Well, I have to say that to me, they both look equally as simple! Especially if we were in the habit of writing $$\frac{+3}{+4}$$ which we don't so I'll just have to go along with what's expected!