I have 4 sinks a bathtub a separate shower and two toilets. I have sewer backup but only in one sink, the kitchen. Why would only one sino be effected? And why a kitcken not a bathroom where it would make more sense?

And why a kitcken not a bathroom
Because the kitchen sink drain line is blocked before it reaches the main line.

Remove the kitchen trap and snake the line. Be thankful its not the main line and everything blocked.

Each plumbing fixture has its own drain pipe until it merges with other drain pipes and into a common sewer pipe. As Harold said, your kitchen drain is blocked in this individual section of drain pipe, before it merges with other drains.

Milo