You may as well solve the system. Multiply first equation by -2:
Add the two equations to get . Clearly, x is not an integer so we're done.
I'm to show that the system of equations below has no integer solutions.
Reducing to modulo 2, we get
This means that must be odd and the sum odd as well. A sum can only be odd if one number is odd and the other odd, thus must be an even number. I haven't seemingly arrived at a condraction, I am still however doing something wrong because the equation in fact doesn't have any integer solutions. Help anybody?
My bad I forgot to write that it needs to be done using modular arithmetic. Considering that I was attempting to do it using modular arithmetic and I posted it under 'Number Theory', it's surprises that it wasn't evident how it's meant to be solved, but I digress.
So, how would we move on from: