Understanding First Order Filters

While sorting through some very old papers I came across a solution to an interesting problem that I I struggled with when I was learning DSP. I have no idea where the original problem came from so I've replicated it here, as best I can remember, along with the solution:

The following first order direct form II filter :

                 w(n)
x(n) -->+-------------------+-->y(n)
        ^         |         ^
        |       +----+      |
        |       |z^-1|      |
        |       +----+      |
        |         |         |
        |         v         |
        ----*-----------*----
           a1  w(n-1)  b1

Is defined by the following equations:

y(n) = w(n) + b1.w(n-1)     (1)

w(n) = x(n) + a1.w(n-1)     (2)

Question: Show the difference equation in terms of y and x ?

Hint: Rearranging to a direct form I filter structure will help.

Solution

Diagramatically

The original system is a Linear Time Invariant (LTI) system so the feedforward and feedback sections can be swapped without changing the system response:

x(n) -------------+-------------->y(n)
         |        ^        |
       +----+     |      +----+
       |z^-1|     |      |z^-1|
       +----+     |      +----+
         |        |        |
         v        |        v
         ----*----+----*----
            b1        a1

Hence:

y(n) = x(n) + b1.x(n-1) + a1.y(n-1)

Mathematically

From (2):

w(n-1) = x(n-1) + a1.w(n-2)     (3)

Substituting (2) and (3) into (1), to compute the output:

y(n) = x(n) + a1.w(n-1) + b1.[x(n-1) + a1.(w(n-2)]     (4)

Rearranging to combine w terms:

y(n) = x(n) + b1.x(n-1) + a1.[w(n-1) + b1.w(n-2)]     (5)

From (1):     y(n-1) = w(n-1) + b1.w(n-2)     (6)

Substituting (6) into (5) gives:

y(n) = x(n) + b1 x(n-1) + a1 y(n-1)

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