‘And he arose from there and went away to the borders of Tyre and Sidon. And he went into a house and would have no one know it. But he could not be hidden, for immediately a woman whose little daughter had an unclean spirit, having heard of him, came and fell down at his feet.'
‘From there.' A general statement meaning ‘from where He was' i.e. in context from Gennesaret - Mark 6:53 (or from the house - Mark 7:17). But there is no indication of how much time had elapsed. It is significant that Mark puts this account immediately after Jesus' statement about nothing from without being able to defile a man. That was a first move necessary for welcoming Gentiles.
‘The borders of Tyre.' In the plural the word can also mean ‘region'. He actually entered the region of Tyre (not Tyre itself). Some good authorities add ‘and Sidon'. Either way the thought is merely that he crossed the border into that region, not that he visited those towns. There is no suggestion anywhere that He entered a town until He reached Bethsaida in Decapolis, and in general he seems to have excluded the idea.
‘Into a house.' Jesus was given a welcome and hospitality, presumably by a Jew who lived in the region (there were many Jews in the area), and His wish was for complete privacy. He did not want His presence to be generally known. It would seem that His main purpose in being here was to have time for rest and recuperation.
No mention is made of the disciples by Mark, although they are mentioned by Matthew. But He was too well known for secrecy to be possible (‘He could not be hidden' - compare Mark 3:8) and word had clearly got around that the new Jewish prophet was in the area and was staying at this house. For within a short time a woman whose daughter was ‘possessed' sought Him out and fell before Him in supplication, an action acknowledging her recognition that He was a man of God.
Matthew lets us know that she did not come to the house but waited until Jesus and His disciples went out for a walk. For a woman, and a Gentile one at that, to come to Jesus in the house would have been heavily frowned on. It would have been seen as bad enough that He spoke to her outside (but Jesus did not feel bound by such prejudices. Compare the Samaritan woman in John 4).