System of equations, how to do this

Question

This is for a physics class, but I think the question is mathematical in nature. We have the following equations:

$$ \mu (c-c') = -m(v-v')$$

$$ \mu (c^2 - c'^2) = -m(v^2-v'^2) $$

$$ c+ c' = v + v'$$

We want to obtain $$ c' = 2v-c + \dfrac{2\mu}{m+ \mu}(c-v)$$

I didn't know how to do this, so I looked at the solution and I understand every step except the first one:

$$ \mu(c-c') = m(c+c' - 2v) $$

What is the logic behind this?

Answer 1

$-m(v-v') = m(v'-v) = m((v+v')-2v) = m((c+c')-2v)$

Answer 2

Pretty simple! From your third equation $c+c'-v=v'$.

Now $-m(v-v')=m(v'-v)=m(c+c'-v-v)=m(c+c'-2v)$