Ok, I'll do this out in full too, and I might do it two ways again I'll do it the way you're having a problem with first

Method 1)

Start with 1.9 = 1.9.

You can split the integer and non-integer part one side as follows:

1.9 = 1 + 0.9

Now we need to do something about 0.9. Let 0.9 = X. X = 0.9 at this point in time only, and nothing else. We can rewrite that equation as

1.9 = 1 + X

This is Equation A.

Now, moving back to our definition of X, we have

X = 0.9

Multiply both sides by 10:

10X = 9.9.

Subtract X from both sides:

10X - X = 9.9 - X

Simplify the LHS:

9X = 9.9 - X

Since X = 0.9 at this point in time only, and nothing else, we substitute this value in the RHS:

9X = 9.9 - 0.9

Simplify the RHS:

9X = 9
X = 1

Now we're allowed to use X = 1 because we've shown it. We can stick this back in Equation A and get

1.9 = 1 + 1
1.9 = 2 ∎


Method 2)

We've got 1.9. Let's call this Y. Y = 1.9. Y is nothing else at this point in time. Y is only 1.9 at the moment.

Y = 1.9

Multiply both sides by 10:

10Y = 19.9

since when multiply by the base we're working in (10), you just shift the (decimal) point.

Subtract Y form both sides:

10Y - Y = 19.9 - Y
9Y = 19.9 - Y

Now Y = 1.9. Y is nothing else at this point in time. Y is only 1.9 at the moment. So we can only substitute this value in the RHS.

9Y = 19.9 - 1.9

Simplify the RHS:

9Y = 18

Divide both sides by 9:

Y = 2

Since Y = 1.9 though, then

2 = 1.9 ∎