Had a professional plumber make original install 9 yrs. Ago. Never had a problem before. Notice now that when bathtub is draining, toilet starts to gurgle and water goes out of bowl. Is this a sign of a clog? Or should I be thinking of something else?

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when bathtub is draining, toilet starts to gurgle and water goes out of bowl

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This tells me that you have a blocked vent. The :gurgle is the tub venting,(pulling air) through the toilet bowl.
The solution would be to snake from the lavatory roof vent. Putt out enough snake to reach the base and 20 feet more.

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washing machine wastewater backups into bathtub drain?

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And that tells me that you have a blockage downstream from the bathtub. You may catch both problems by snaking the roof vent but if snaking the roof vent doesn't get the blockage remove the tub overflow and snake from there,(see image) Good luck, Tom