this can be demonstrated in many ways.
Normally I wiould show you with number patterns.
Today I'll try something different - logic 
Say you borrow 'parcel' of $3 from your mum. You borrow 5 parcels of $3 altogether.
If you owe someone money than this is a negative amount that you have (really don't have)
So you now have 5*-$3 = -$15
Now you want to get rid of this debt you want to SUBTRACT this debt (get rid of it). Then you will have nothing (and owe nothing)
so -15-(-15)=-15+15=0
(5*-3)-5*-3=0
but 20-2 is the same as 20+(-2)
that is
(5*-3)+(-5*-3)=0
-15+ [ ]=0
well 15 has to go in the box so -5*-3=+15
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I expect that is about as clear as mud. LOL
I don't think i will try to explain it that way again. 
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I have answered differently in the next post.
You can comment on my answer if you are there. Think about it a little first though. 