Problem $2$:A company produces three types of products: A$,$ B$,$ and C$.$ The production of these

products involves three different machines: M$1,$ M$2,$ and M$3.$ The usage of each machine for each

product $($in hours per unit of product$)$ is given below:

Machine M$1$: $2$ hours for product A$,4$ hours for product B$,$ and $-6$ hours for product C$.$

Machine M$2$: $3$ hours for product A$,-1$ hour for product B$,$ and $2$ hours for product C$.$

Machine M$3$: $-1$ hour for product A$,5$ hours for product B$,$ and $3$ hours for product C$.$

The total available hours for each machine per week are as follows:

Machine M$1$: $7$ hours

Machine M$2$: $-3$ hours

Machine M$3$: $8$ hours

Determine the number of units of products A$,$ B$,$ and $C$ that the company can produce in a

week to fully utilize the available machine hours. Solve by Cramer's rule to find the values of $A,$

$B,$ and $C.($Shows the steps and the work$)(10$ points$)$