I solved the problem by process of elimination. (no article)

I used the process of elimination to solve the problen. (the definite article "the")

Do these sentences sound natural?

Question

I solved the problem by process of elimination. (no article)

I used the process of elimination to solve the problen. (the definite article "the")

Do these sentences sound natural?

Answer

Yes, both sentences sound natural.

Explanation:
"by process of elimination"

The phrase is used without an article as a fixed expression. It emphasizes the method used.
"the process of elimination"

The definite article "the" is used to refer to a specific, well-known method (the process of elimination).
Both forms are correct, just slightly different in tone.

Answer

"A process of elimination"  works better in both sentences.

Answer

yes! these sentences sound very natural and are most likely used often.