Let us take an example where $a+b$=3, $a-b$=1, and $ab=2$. Solving the first two equations give the values of $a$ and $b$ as 2 and 1 respectively and putting these values into last equation yields an identity. So, in no way it means here that the system will have infinite solution. So, the proposition in the question is false.
But I found the proposition to be written in my Maths notes. So, I wanted to have your views on this. May be I've miswritten something in my notes. Am I right?