Normally bubbles should not appear in your toilet's bowl, especially when you drain waste water out of another plumbing fixture such as the washing machine. Bubbles in the toilet's bowl indicate a potentially serious problem with the plumbing, which you should look into as soon as possible to avoid more serious or expensive repairs later.
Partial Clogs
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A partial clog in the toilet's drain line or the larger drainpipe that the toilet and washing machine both feed into may lead to bubbles appearing in the toilet's bowl. A partial clog still allows some water to flow through the drainpipe, but the clog traps air behind it. As you flush waste water down the drain, the disturbance leads to some of the air behind the partial clog escaping in the form of small bubbles. Air that flows down the drainpipes along with the waste water helps replenish the air supply trapped behind the partial clog, meaning the bubbles may continue to appear until the clog either comes loose or grows to a complete blockage.
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Snaking the Drains
To eliminate a clog that sits deep in the drain line, you must unseat your toilet from the floor. Always empty all of the water out of the toilet before moving it, or else you will have a large mess on your hands. Also, be careful when moving the toilet, since it can easily break if you set it down too quickly. Feeding a drain snake down the toilet's drainpipe in the floor gives you the best chance of reaching the larger drain pipe and any potential partial clogs in the larger pipe, since the toilet's drainpipe sits flush with the floor.
Venting Problems
If a partial clog is not the cause of bubbles forming in the toilet's bowl when you run the washing machine, the cause may be from poor venting of the larger drainpipe. A vent pipe connects to the larger drainpipe, venting out sewer gases in the larger pipe so the gases do not try to escape through the plumbing fixtures and into the house. Also, the vent pipes introduce outside air, helping replace air that washes down the drain pipes along with waste water.
Remove Vent Clogs
You must go onto your roof to check for clogs in your vent pipes, so choose a day when the weather does not create a hazard for being on your roof. Sometimes clogs come in the form of a bird's nest or other object on the top of your vent pipe, which you can remove easily. Use a flashlight to look down your vent pipes for clogs inside. If you cannot reach a clog in your vent pipes, turn on a garden hose to knock the clog free.