# How to use parameterized solution to find solutions?

An electronics company produces three models of stereo speakers, models A, B and C, and can deliver them by station wagon, truck, or van. A station wagon holds 3 boxes of model A, 3 boxes of model B and 2 boxes of model C. A truck holds 7 boxes of model A, 6 boxes of model B and 4 boxes of model C. A van holds 4 boxes of model A, 3 boxes of model B and 2 boxes of model C. If 23 boxes of model A, 21 boxes of model B and 14 boxes of model C are to be delivered and the owner wants each vehicle to operate at full capacity, the parameterized solution to the resulting system of equations is: (z+3,-z+2,Z). for (station wagons, trucks, vans). Use the parameterized solution to find the possible configurations of vehicles.

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Please don't post questions in the imperative mode, as if giving orders or giving out an assignment. Also, please don't use the title as an integral part of the question, use the body of the question for that. It helps a lot if you say where you are stuck and why. –  Arturo Magidin Feb 7 '11 at 16:29

What the parametrized solution tells you is that $$\left(\begin{array}{c} \text{Station wagons}\\ \text{Trucks}\\ \text{Vans} \end{array}\right) = \left(\begin{array}{c} 3\\ 2\\ 0 \end{array}\right) + \left(\begin{array}{r} 1\\ -1\\ 1 \end{array}\right)z.$$ So each value of $z$ will give you a configuration. That is, the number of vans will determine how many station wagons, how many trucks, and how many vans you need.
Of course, since you want the answer to give you the number of vehicles, that places some restrictions on $z$: it cannot be, for instance, $\pi$, because that would mean you have $\pi$ vans, and that's going to be a little difficult to arrange (not to mention, to get insurance coverage for). And you cannot have $z=-500$, because that would require the number of trucks to be $-498$, which again seems somewhat difficult to accomplish. So there are only so many values of $z$ for which the mathematical answers actually make sense for the problem as it is to be applied. Figure out what those values are, and then use them to figure out what each of them means in terms of the number of vans, trucks, and station wagon.