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

1. Jun 26, 2012

### Jherek

1. The problem statement, all variables and given/known data

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

2. Relevant equations

3. The attempt at a solution
$$s = \frac{-t-5u-11}{-4u}$$
$$s=\frac{t+5u+11}{4u}$$
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?

2. Jun 26, 2012

### 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.

3. Jun 26, 2012

### 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.

4. Jun 29, 2012

### Staff: Mentor

I suspect that what you were told is that on the right-hand side "we can multiply both "top" and "bottom" by -1". 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)

5. Jun 29, 2012

### SammyS

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

6. Jun 29, 2012

### 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!