A bank teller has a total of $73$ bills in five$-,$ ten$-,$ and twenty$-$dollar denominations. The total value of the money is $$970.$

$($a$)$ Find the total number of solutions.

$($b$)$ Find the solution with the smallest number of five$-$dollar bills.

$($c$)$ Find the solution with the largest number of five$-$dollar bills.

$($a$)$ Let $x$ be the number of five$-$dollar bills, $y$ be the number of ten$-$dollar bills, and $z$ be the number of twenty$-$dollar bills. Translate the given information into two equations,

using $x,y,$ and $z$ as the variables.

$x+y+z=73$

$=970$

$($Do not factor. Use integers or decimals for any numbers in the expressions.$)$

There are a total of solutions.

$($b$)$ The solution with the smallest number of five$-$dollar bills is when the bank teller has five$($s$),$ ten$($s$),$ and twenty$($twenties$).$

$($c$)$ The solution with the largest number of five$-$dollar bills is when the bank teller has five$($s$),$ ten$($s$),$ and twenty$($twenties$).$