Question

Write the augmented matrix for the following problem. A cub scout is selling bags of popcorn. The small bags cost $3, the medium bag cost $5, and the large cost $7. He has sold 15 bags for total sales of $77. He sold 2 more medium bags than small bags. How many of each type of bag did he sell

Answer 1

Answer:

The augmented matrix is:

$\left[\begin{array}{cccc}1&1&1&15\\-1&1&0&2\\3&5&7&77\end{array}\right]$

Step-by-step explanation:

Ntice there are three unknowns:

Number of small bags sold , let's name it S

Number of medium bags sold , let's name this M

Number of large bags sold, let's name this L.

We need now to start by writing the three equations that relate these three unknowns

1) Total number of bags sold: 15

S + M + L = 15   equation (1)

2) Two more medium bags were sold than small bags

M = 2 + S  which can also be written as: M - S = 2   equation (2)

(grouping all unknowns on the left and pure numerical on the right of the equal sign)

3) The total sales is $77, so we create a value equation using the value of each item times the nunber of those sold:

3 S + 5 M + 7 L = 77  equation (3)

Now we write the augmented matrix for these three equations using the first column for the variable S (number of small bags), the second colum for the variable M (number of medium bags), the third column for L (number of large bags), and the 4th column for the strictly numerical terms that we located on the right hand side of the equal signs in all three equations:

$\left[\begin{array}{cccc}1&1&1&15\\-1&1&0&2\\3&5&7&77\end{array}\right]$

The solving of this augmented matrix via RREF calculations results in:

S = 4, M = 6, and L = 5