Algebra method explained:

Since we do not know exactly how many $2 and $10 notes there are, we use an unknown called 'x' to create our equations.

Since the ratios of $2 to $10 notes is 5:2, we can say that:

Number of $2 notes = 5x (5 times unknown x)
Number of $10 notes = 2x (2 times unknown x)

Notice that whatever number x is, the ratio will always be 5 to 2.

When the exchange happens, the number of $2 notes increase by 10 and the number of $10 notes decrease by 2.

So we can write the new number of notes as:

New number of $2 notes = 5x + 10
New number of $10 notes = 2x - 2


And we know that the ratio of the new notes is 5:1, meaning that we can express their ratios as:

(5x + 10)/(2x - 2) = 5/1

Apply your cross multiplication, you get...

10x - 10 = 5x + 10
5x = 20
x = 4

We recall that the original number of $10 notes was represented by the term 2x, so that will be 8.

Hope this makes it clearer :)