A car rental agency rent compact, midsize, and luxury cars. Its goal is to purchase 48 cars for a total of $932,000 and to earn a daily rental of $ 1,265 from all the cars. The compact cars cost $13,000 each earn $ 20 per day in rental, the midsize cars cost $ 25,000 each earn $ 27 per day, and the luxury cars cost $ 29,000 each and earn $ 45 per day. Your task will be to find the number of each type of car the agency should purchase to meet its goal.

The set of relationships can be formulated as three equations in three unknowns. These can be written as an augmented matrix and solved using a calculator.

Let x, y, z represent the numbers of compact, midsize, and luxury cars, respectively. Then the relations are ...

x + y + z = 48 . . . . . . total number of cars purchased

13x +25y +39z = 932 . . . . . total cost in thousands

20x +27y +45z = 1265 . . . . daily rental revenue

The augmented matrix of coefficients looks like ...

[tex]\left[\begin{array}{ccc|c}1&1&1&48\\13&25&39&932\\20&27&45&1265\end{array}\right][/tex]

And its solution is ...

(x, y, z) = (25, 15, 8)

The agency should purchase 25 compacts, 15 midsize, and 8 luxury cars.