When can I be confident that I can cancel out variables?

when you get an equation like 3w^2 = 6w, you might be tempted to cancel the w's, and it's perfectly cool to do so, but remember WHAT cancelling means:

when you cancel the w's here what you are implicitly doing is DIVIDING BOTH SIDES BY "w"

seems harmless cuz there's a W on each side right? except for one thing, if the W happens to be zero, you're dividing by 0, which is one of math's cardinal sins!

so when you do this operation, keep in mind the fact that W might actually also be 0 (which you can easily test: plug in 0 into W, and if the equation works, the 0 is at least one of the answers). If this is the case, you can still simplify, but you need to keep the case where w=0 in your back pocket in case you need to use it, and if you don't need to use it, no harm done: your equation is simplified

hope this helps, but don't be shy if it doesn't!