You're just saying that a body is required to have breath; one can't live without the other. I agree with that much. Works can't live without faith, hence the term "dead works." Dead works are things done without faith and they are sin. Remember, it's impossible to please God without faith. Whatever is not from faith is sin.
Keeping James 2 in mind, where Abraham was justified by his faith and his works, Ephesians 2 is clear we are saved by grace through faith and works,
not of [dead] works.
Faith + good works are required:
Ephesians 2
8For it is by grace you have been saved, through faith—and this is not from yourselves, it is the gift of God—
9not by works, so that no one can boast.
10For we are God’s handiwork, created in Christ Jesus to do good works, which God prepared in advance for us to do.