**Answer:**

We have 4 solutions:

**No** 10-foot truck, **10** 14-foot trucks, and **15** 24-foot trucks**2** 10-foot trucks, **7 ** 14-foot trucks, and **16** 24-foot trucks**4** 10-foot trucks, **4** 14-foot trucks, and **17** 24-foot trucks**6** 10-foot trucks, **1 ** 14-foot trucks, and **18** 24-foot trucks

**Step-by-step explanation:**

Let the number of 10-foot truck with a capacity of 350 cubic feet purchased=a

Let the number of 14-foot truck with a capacity of 700 cubic feet purchased=b

Let the number of 24-foot truck with a capacity of 1,400 cubic feet purchased=c

The company wants to purchase 25 trucks, therefore.

Furthermore, the combined capacity of the trucks is 28,000 cubic feet.

Since the number of equations is less than the number of variables, you can not use a matrix equation to solve this problem. The solution is most easily found using an augmented matrix.

The augmented matrix is presented below:

Using the calculator, the reduced row echelon form is:

where

a- 2c=-30 means a =2c-30

b+3c=55 means b= 55-3c

We alter the value of c as long as neither a nor b becomes negative. Suitable values for c are 15, 16, 17, and 18:

We can easily verify that, for each solution, the number of trucks adds up to 25 and the fleet capacity is 28,000 cubic feet.

We therefore have 4 solutions:

**No** 10-foot truck, **10** 14-foot trucks, and **15** 24-foot trucks**2** 10-foot trucks, **7 ** 14-foot trucks, and **16** 24-foot trucks**4** 10-foot trucks, **4** 14-foot trucks, and **17** 24-foot trucks**6** 10-foot trucks, **1 ** 14-foot trucks, and **18** 24-foot trucks