finite math word problems. Applications of systems of linear equations

Question

A Starbucks store sells coffee in three sizes: a12-oz tall, a 16-oz grande, and a 20-oz venti. A tall coffee sells for $1.75, a grande coffee for 1.95, and a venti coffee for 2.25. One morning, brandie served 50'coffees for a total of 98.70. She made the coffee in 80 oz batches and used exactly 10 of the batches during the morning. How many of each size did she sell? Write the three equations you would use to solve. Using Gauss-Jordan elimination, solve the system.

Answer 1

Hi JZ:

 

First let us assign some variables / letters to the unknown quantities:

 

T = number of Tall coffees

G = number of Grande coffees

V = number of Venti coffees.

 

Now let us write the equations representing:

 

(1) The total number of coffees:

T + G + V = 50

 

(2) The total cost of the coffees (in dollars):

1.75*T + 1.95*G + 2.25*V = 98.70

 

(3) The total number of ounces:

12*T + 16*G + 20*V = 80*10 = 800

 

Now you have 3 linear equations of 3 unknown, which can be written in a matricial form (kind of difficult to write here, but I will try):

 

 |  1       1     1     |         |  T  |        |  50   |

{ 1.75  1.95  2.25   }   x  {   G   }  =  {  98.7  }

 |  12     16    20   |         |  V  |        | 800  |

 

3 x 3 Matrix * Unknown Vector = Known vector

 

Now you can apply the Gauss-Jordan elimination to this 3 x 3 linear system, and the solution is:

 

T = 12, G = 26, V = 12.

 

Hope this helps!

 

  Jean-Michel