Suppose $m,n,a,a',b,b' \in \mathbb Z$ and $m \ne n$ and the following is true: $$ma+nb=ma'+nb'$$ Then would it be correct to conclude $a=a'$ and $b=b'$? If yes, then prove it. Also, if possible, which assumptions can I remove in my hypothesis such that the conclusion still remains true?
Edit: If this is not true, can I add some condition in my hypothesis to make it true?
No, it is not correct, since for each integer $k$,$$m(a+kn)+n(b-km)=ma+nb.$$