" to her husband, who was with her, and he ate"

So, despite what Jeremy Crews says, this is not a feminist blog.

We often attribute Eve with the first sin, rightfully so I suppose as she did take the first bite of the apple. She succombed to temptation and listened to the serpent.

However, we see from the text, that "she gave some to her husband, who was with her, and he ate." So here enters my question, is she to blame? If the husband, as the spiritual head of the family was with her, why did he not stop her? Furthermore, as this was before the fall, were they equal, or does the fact that Eve was taken from Adam's hip communicate that women were to submit to men in the beginning?
Eve was, "the bone of my (Adam's) bones and the flesh of my flesh. She will be called woman, for she was taken out of man." Genesis 1:23
This leads me to believe that women were always meant to submit to men. We were taken from men, so should they not lead us? He created man in His own image, and us from man. So wherein lies the balance of submission vs. well...a doormat?
It lies in Christ.
When Christ leads the man and woman, and the man leads the woman as well, it becomes more than, "honey will you make me a sandwich?" It becomes a picture of Christ and the church. Christ leads, protects, and provides (in addition to so much more) and the church submits out of love and obedience.
So, circling back to Genesis 3, why didn't Adam protect Eve from the serpent? Why didn't he protect her purity to the best of his ability? So is Adam just as much to blame?