# How do you solve this Linear Algebra Problem?

Use what you know about linear algebra and matrices to solve the following word problem: A rental

company has a

eet of 12 vans which can carry a total of 220 passengers. The vans come in three sizes

(A, B, and C) according to the number of passengers they can hold: 10, 15 and 20, respectively. How

many of each van must be rented so as to carry a total of 220 people? List all possible solutions.

LIN Unit 1 Test Page 2 of

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From this information I can only get 2 appropriate equations:

a + b + c = 12

10a + 15b + 20c = 220

If you were to make it a matrix:

[ 1 1 1] = 

[ 10 15 20] = 

It's not a square matrix so there is no actual solution

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• You have the matrix

[1 1 1 12 ]

[10 15 20 220]

or

[ 1 1 1 12]

[ 0 5 10 100 ]

after using GAUSS elimination

that means 5B +10C = 100

or B +2C = 20

let C= Z (an integer)

then B = 20 -2Z

and A = Z - 8

since A >0 then Z> 8

since B > 0 then Z <10

Z > 8 and Z <10 then Z must be 9

then C = 9 , A = 1 , B = 2

check :

A+ B +C =9 +1 + 2 =12

and 10A + 15B + 20C = 10 + 15(2) +20(9)

= 10+30 +180 = 220

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