Difference Equations - Reduction of order

Question

I am asked to change the order of the following to a first order:$$Y_{t+2}-3Y_{t+1}+4Y_t=2$$ The approach I took was to create another equation and got the following system. $$\begin{cases} Y_{t+2}=12-3Y_{t+1}+2Z_{t+1} \\ Z_{t+2}=Y_{t+1} \end{cases}$$Is this correct?

Answer 1

Let $x_t=Y_{t}-1$. Then you have $$ x_{t+2}-3x_{t+1}+4x_t=0. $$ Let $z_t=x_{t+1}+x_t$ and hence you have $$ z_{t+1}=4z_{t} $$ which is first order.