This is interesting @manuelhooks. Just that aspect of the factors of 10 split into two multiplication steps and also thinking of a number when subtracted from the results produces 7, this sets the stage on.
As you have rightly said, this is a trick and Not magic.
So let me play with a number that I am thinking about now;
Let the number be z, I will multiply it by 5:
Step1 =5×z =5z
I multiply my product by 2:
Step2 = 2× 5z =10z
I finally subtract 3 from the result of step2
Step3 = (10z-3) ; 7 is expected to be in the answer because, there is 10 & 3.

Here I go, if my z = 6.
Then Step1: 6×5 =30
Step2: 2×30 =60
Step3: 60-3 =57.
An Interesting algebra!
Let open our minds to learn tricks with numbers. It's not a magic.