Who's fault is it anyway?

As we learned in class, Adam only ate the fruit because Chava told him it was okay to eat. We learned that the snake tricked Chava, who then convinced Adam to eat. I have heard elsewhere that Hashem told Adam 'don't eat from this tree' and Adam told Chava 'don't touch the tree or eat from it'.
If this is true, was it really Chava's fault that she ate the fruit? The snake pushed Chava into the tree. When nothing happened to her, she began to doubt what she was told, but should she have told Adam before she ate the fruit?
In my opinion, Chava is partly to blame. Despite the warning, when the snake pushed her into the tree, she then proceded to eat from the fruit. Adam is also to be blamed. Instead of adding to what he was commanded, he could have told Chava what Hashem had said, and warned her not to touch the tree because there was a possibility it would tempt her.