Question

The​ U-Drive Rent-A-Truck company plans to spend ​$8 million on 280 new vehicles. Each commercial van will cost ​$25 comma 000​, each small truck ​$30 comma 000​, and each large truck  ​$40 comma 000. Past experience shows that they need twice as many vans as small trucks. How many of each type of vehicle can they​ buy? They can buy nothing ​vans, nothing small​ trucks, and nothing large trucks.

Can you demonstrate the Guass Jordan method for solving this?

Answer 1

Let the number of commercial vans, small trucks and large trucks purchased by the company be x,y,z respectively.

Since the ​ U-Drive Rent-A-Truck company plans to buy 280 new vehicles, we have x+y +z = 280….(1

Further, since the company needs twice as many vans as small trucks, we have x =2y or, x-2y = 0…(2).

Also, the total amount to be spent by the company is $ 8000000, so that 25000x+30000y +40000z = 8000000 or, 5x+6y+8z = 1600…(3).

The augmented matrix of this linear system is A (say) =

1	1	1	280
1	-2	0	0
5	6	8	1600

To solve the above linear system, we will reduce A tio its RREF as under:

Add -1 times the 1st row to the 2nd row

Add -5 times the 1st row to the 3rd row

Multiply the 2nd row by -1/3

Add -1 times the 2nd row to the 3rd row              

Multiply the 3rd row by 3/8

Add -1/3 times the 3rd row to the 2nd row

Add -1 times the 3rd row to the 1st row

Add -1 times the 2nd row to the 1st row

Then the RREF of A is

1	0	0	160
0	1	0	80
0	0	1	40

Thus, the company would buy 160 commercial vans, 80 small trucks and 40 large trucks.