If 7+3=10, why is 77+33 not equal to 100?
Because your comparison is not valid.

First you state that 7+3=10. That is ok.

Then you multiply 7 and 3 each by 11 to get 77 and 33.

Then you try and make that sum equivalent to 10 x 10, which is not valid.

if you had multiplied the other side by 11 it would have worked, because

77 + 33 = 110 (i.e. 10 X 11)

or if you had multiplied by 10 to start with it would have worked, because

70 + 30 = 100.

But you started with an erroneous assumption. You need to be careful with that. It is always a good idea to take a step back and ask yourself “What am I assuming in step 2? Is it valid? Is it correct?“

Were you just “doubling the digits”? I.e 7 becomes 77 or 777? If so then why didn’t you double the digits on the other side of the equals (10 becomes 110 or 1110)?

because 6 + 4 also equals 10 and 66 + 44 also equals 110 and 666 + 444 also equals 1110 (as does 777 + 333)

Here is something interesting for you

we now know that

77+33=110

and that

66+44=110

if we subtract these two equations from each other we should still be equal. That is

77+33=110

−66+44=110

(77−66)+(33−44)=(110−110)

11+(−11)=0