Suppose you have three bank accounts with different banks. In each month, for Account $1,$ you will pay

$C\%$ of your balance of last month for daily expense, and transfer D$\%$ of your balance of last month to

account $3$; for Account $2,$ you will transfer E$\%$ of your balance of last month to Account $1,$ transfer F$\%$ of

your balance of last month to Account $3,$ and save your salary in this account $($Account $2)$; and for Account

$3,$ you will pay the mortgage from this account. $(C$ is the last digit of your student $ID$ times $10($if $C=0,$

then change to $C=50$; if $C80,$ then change to $C=C-20)$; $D=100-C-20$;$E$ is the second last digit of your

student ID times $10($if $E=0,$ then change to $E=40$; if $E90,$ then change to $E=E-10)$; $F=100-E-10.)$

There are no other transactions. Suppose the state of the system $x(k)=[x_1(k),x_2(k),x_3(k){]}^T$ is the balance of

each account at month $k.$ For each month your salary is $u_1(k)$ and the mortgage you need to pay is $u_2(k),$

which are the input $u(k)=[u_1(k),u_2(k){]}^T.$ And your initial balance of the three accounts are $x_1(0)=32,000,$

$x_2(0)=9,500$ and $x_3(0)=18,000.$

For example:

If a student's ID is $10588945,$ then $C=5^{**}10=50,D=100-50-20=30,E=4**10=40,F=100-40-10=$

$50.$

If a student's $ID$ is $10588998,$ then $C=8**10-20=60,D=100-60-20=20,E=9**10-10=80,F=100-80-$

$10=10.$

If a student's ID is $10588900,$ then $C=50,D=100-50-20=30,E=40,F=100-40-10=50.$

$(1.$a$)$ Please model it as a linear discrete$-$time system.

$(1.$b$)$ What is the solution of the state $x(k)?$