what saves?

In response to the woman washing his feet with her hair, Jesus says : 
Luke 7:50 And he said to the woman, “Your faith has saved you; go in peace.”

To the woman who reached out in the crowd to touch him:
Luke 8:48 And he said to her, “Daughter, your faith has made you well; go in peace.”

To the one returning healed leper:
Luke 17:19 And he said to him, “Rise and go your way; your faith has made you well.”

To the blind beggar:
Luke 18:42 And Jesus said to him, “Recover your sight; your faith has made you well.” 

The same original word that is translated "saved" in the first example is translated "made you well" in the others, suggesting that something more than physical healing is going on in all of them, but even so, these verses can be confusing. Who exactly is doing the healing and/or the saving?

If you view faith as something you can build up in yourself in order to achieve something, or convince yourself of something, then these verses may suggest that these people managed to drum up enough faith in themselves in order to impress Jesus or earn something from him. 

If we understand faith as a natural response to encountering Jesus, this looks a lot different.

All the people have done is believed and received. Their actions in coming to Jesus are a result of the fact they believe he can do something, they believe he can save them. Who saved them? Jesus. Not themselves. “Faith” in itself has not saved them, faith in Jesus has saved them because Jesus saves.