Think of any number you want to replace y. Let's say you pick 3. So y = 3.
Replace every copy of y with that number 3. You will go from y = y+1 to 3 = 3+1
Then simplify the right side and you end up with 3 = 4, which is false. The number 3 is different from the number 4, so they can't be equal. This contradiction is one example (of many) that helps prove y = y+1 is false.
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To prove that y = y+1 is false for every real number, you would subtract y from both sides like so
y = y+1
y-y = y+1-y
0y = 0y+1
0 = 0+1
0 = 1
and we have another contradiction here, so there is no way this equation has a solution. We say that this equation is inconsistent