# Equivalence of difference equation with system of difference equations

I have this system and tried to write down its difference equation as follows

$$y[n] = b_0v[n] + b_1v[n-1]\\ v[n] = x[n] - a_1v[n-1]$$

I was asked to prove that the system can be described by the following difference equation

$$y[n] + a_1y[n − 1] = b_0x[n] + b_1x[n − 1]$$

Just express $$y[n]$$ and $$x[n]$$ in terms of $$v[n-k]$$, $$k=0,1,\ldots$$, and show the equality.
• @Adam: Well, you have two equations involving $y[n]$, $x[n]$, and $v[n-k]$, so you can express $y[n]$ and $x[n]$ in terms of $v[n-k]$ and then show the given equality. – Matt L. Apr 16 '19 at 12:28