The problem:
A travel company books blocks of rooms are different resorts at a huge discount which it then provides to its clients at a markup.
Resort A has 6 suites, 3 deluxe rooms, 9 standard rooms, and 3 economy rooms in each block
Resort B has 6 suites, 4 deluxe, 6 standard, and 4 economy rooms in each.
Resort C has 7 suites, 4 deluxe, 9 standard, and 4 economy rooms in each.
Resort D has 7 suites, 5 deluxe, 4 standard, and 3 economy rooms in each.
If customers need 155 suites, 95 deluxe rooms, 170 standard rooms, and 85 economy rooms, how many blocks from each hotel should the travel company book so that every room is occupied. Is this possible? If so, determine how many ways there are to do it.
This is for a linear algebra class, and we're supposed to use the matrix form of systems of equations to solve it (rref, linear combinations etc).
I tried using $Ax = b$ where $A$ is the matrix representing the number of rooms of each type in each block from each resort, $x$ is the variable number of blocks to book from each resort, and b is the desired number of rooms of each type (155 suites, 95 deluxe rooms, 170 standard rooms, and 85 economy rooms). But I can't figure out how to conclusively answer the question one way or the other. Is there an $x = (a, b, c, d)$ that yields the exact number of each type of rooms required?
In your matrix, use rows to solve for each type of room, and then columns to designate the resort:
$$A=\begin{pmatrix} 6 & 6 & 7 & 7\\ 3 & 4 & 4 & 5\\ 9 & 6 & 9 & 4\\ 3 & 4 & 4 & 3 \end{pmatrix}.$$
Then $b$ will be the number you need of each: $$b=\begin{pmatrix} 155\\95\\170\\85 \end{pmatrix}.$$ The solution $$x=\begin{pmatrix} x_A\\ x_B\\x_C\\x_D \end{pmatrix}$$ will then denote the number of blocks from each resort you need (or possibly several solutions).
So after you've found all possible solutions, look for integer solutions. Solve one of these for integer values, e.g. solve $3x_A+4x_B+4x_C+3x_D=85$ for nonnegative integers $x_A,x_B,x_C,x_B$. This will give an upper bound on the number of integer solutions. Then plug these back into the solution for the system and see if they fit that form.
Either that, or you'll find some subset of the solution set with $x_A,x_B,x_C,x_D$ being nonnegative. Look for integer solutions in here.
NOTE: There may be an easier way to do this last part, but alas I have not studied integer programming.