$(1$ point$)$ Liam opens a bank account with an initial balance of $1000$ dollars. Let b$($t$)$b$($t$)$ be the balance in the account at time tt$.$ Thus b$(0)=1000$b$(0)=1000.$ The bank is paying interest at a continuous rate of $5\%$ per year. Liam makes deposits into the account at a continuous rate of s$($t$)$s$($t$)$ dollars per year. Suppose that s$(0)=800$s$(0)=800$ and that s$($t$)$s$($t$)$ is increasing at a continuous rate of $2\%$ per year $($Liam can save more as his income goes up over time$).$

$($a$)$ Set up a linear system of the form

dbdtdsdt$==$m$11$b$+$m$12$s$,$m$21$b$+$m$22$s$.$dbdt$=$m$11$b$+$m$12$s$,$dsdt$=$m$21$b$+$m$22$s$.$

m$11=$m$11=,$

m$12=$m$12=,$

m$21=$m$21=,$

m$22=$m$22=.$

$($b$)$ Find b$($t$)$b$($t$)$ and s$($t$)$s$($t$).$

b$($t$)=$b$($t$)=,$

s$($t$)=$s$($t$)=.$