Zero knowledge proof with sum All zero-knowledge protocols I have read so far for discrete log works like this:

1) Prover generate a random number $r$, creates a commitment $t=g^r$ and sends $t$ to the verifier.

2) Verifier generates a random challenge $c$ and sends it to the prover.

3) Prover creates a response $s=r+x*c$ and sends it to the verifier 

My question is: is the security affected in any way if, in step 3, the prover responds with $s=r+x+c$?

I don't understand why multiplication $x*c$ is used instead of sum $x+c$.
Hint: what happens if the prover, instead of sending $g^r$ for a random $r$ that he knows in the first round, sends instead $g^{-x}\cdot g^r$ for a random $r$ that he knows?