CT 6 Chapter 12 Example 12.3 and the Core Reading Question on next page

[A L]

I keep getting confused when i go back and look at these.
Can you explain how it goes from the first line:
xt - 5xt-1 + 7xt-2 - 3xt-3
to the second line in brackets please in simpler terms?
And also for the question at the top of the next page please.

 

[Bharti Singla]

Since we need to express the given expression in terms of difference operator that is ∆Xt= Xt - Xt-1,
We need same coefficients of (Xt, Xt-1), (Xt-1,Xt-2) and (Xt-2, Xt-3).

In this example, we need same coefficient of Xt-1 as that of Xt (which is 1). So, we can split 5Xt-1 into Xt-1 and 4Xt-1.
Now, Xt - Xt-1 will can be written as ∆Xt and we are left with 4Xt-1 part.

Again we need same coefficients for Xt-1 and Xt-2. Xt-1 has coefficient 4(from above result) and we can split 7Xt-2 into 4Xt-2 and 3Xt-2.
Now, 4Xt-1 - 4Xt-2 would become 4∆Xt-1.

We are left with 3Xt-2(from above result) and 3Xt-3 now, which can be written as 3∆Xt-2.

Hope this helps!