Finding optimum values for 2 unknown variables ![enter image description here]( How do I find the maximum numbers of rooms which can be booked for $100 and $150 rooms? And what topics do this kind of problems belong to?

First of all you should define variables: $x$:= number of 2-person rooms, $y$:= number of 4-person rooms. Then use the two conditions to get 2 equations.

The total budget (\$9,600) must be equal to the expenses: $$100x+150y=9600 \quad $$

The sum of persons in 2-person rooms and in 4-person rooms must be equal to 216:

$$2x+4y=216 \quad $$

Now use the two equations to calculate the values for $x$ and $y$. Solving the second equation for x:

$2x=216-4y$

$x=108-2y \quad (\color{blue}*)$

Now we can insert the term for x into the first equation.

$100\cdot (108-2y)+150y=9600$

$10800-200y+150y=9600$

$10800-50y=9600\quad |+50y $

$10800=9600+50y \quad |-9600$

$1200=50y$

$24=y$

Finally use $(\color{blue}*)$ to calculate the value of $x$.

**Remark**

There is no optimum, since due to the restrictions nothing can be optimized. The number of employees are given and the budget is given.