A Syrophoenician woman had a daughter who was possessed by a demon. The woman sought Jesus' aid for her daughter, and Jesus told her that God sent Him to save the Jews. When the woman continued to beg, Jesus told her that food meant children (Jews) were not meant for house pets (Gentiles). Notably, Jesus called her a pet dog instead of a street dog. As the Amplified Bible notes:
Jews used kuon (dog) as a derogatory term referring to Gentiles. This dog (kuon) was a despised, filthy, homeless street scavenger. When speaking with this woman, Jesus uses a word for “dog” (kunarion) that refers to a household pet. The use of the word kunariois by both Jesus and the woman reflects the tenderness and spiritual depth of this exchange. More importantly, it foreshadows the fact that Gentile believers would not be spiritually homeless, but would also be welcomed into God’s household as His children. The gracious response of the woman recorded in v 27 confirms that on some level she understands this.— Footnote for Matthew 15:26 (AMP)
The woman then told Jesus that pet dogs often eat crumbs that the children leave behind. Due to her answer and faith, Jesus told her that her daughter was healed. The woman returned home to find her daughter healed and the demon gone.
After leaving there, Jesus withdrew to the district of Tyre and Sidon. And a Canaanite woman from that district came out and began to cry out [urgently], saying,Have mercy on me, O Lord, Son of David (Messiah); my daughter is cruelly possessed by a demon.But He did not say a word in answer to her. And His disciples came and asked Him [repeatedly],Send her away, because she keeps shouting out after us.He answered,I was commissioned by God and sent only to the lost sheep of the house of Israel.But she came and began to kneel down before Him, saying,Lord, help me!And He replied,It is not good (appropriate, fair) to take the children’s bread and throw it to the pet dogs.She said,Yes, Lord; but even the pet dogs eat the crumbs that fall from their [young] masters’ table.Then Jesus answered her,Woman, your faith [your personal trust and confidence in My power] is great; it will be done for you as you wish.And her daughter was healed from that moment.— Matthew 15:21–15:28 (AMP)
Jesus got up and left there and went to the region of Tyre [and Sidon, the coastal area of Phoenicia]. He entered a house and did not want anyone to know about it; but it was impossible for Him to be hidden [from the public]. Instead, after hearing about Him, a woman whose little daughter had an unclean spirit immediately came and fell at His feet. Now the woman was a Gentile (Greek), a Syrophoenician by nationality. And she kept pleading with Him to drive the demon out of her daughter. He was saying to her,First let the children [of Israel] be fed, for it is not right to take the children’s bread and throw it to the pet dogs (non-Jews).But she replied,Yes, Lord, but even the pet dogs under the table eat the children’s crumbs.And He said to her,Because of this answer [reflecting your humility and faith], go [knowing that your request is granted]; the demon has left your daughter [permanently].And returning to her home, she found the child lying on the couch [relaxed and resting], the demon having gone.— Mark 7:24–7:30 (AMP)