Question

Peter processed two catering orders at the sandwich shop where he works. The first order was for 7 trays of club sandwiches and 10 trays of vegetarian sandwiches. That order cost $149. The second order, which cost $10, was for 1 tray of vegetarian sandwiches.

This situation is modeled by the matrix equation shown.
[7   10]  [c]    [149]
[0    1 ]  [v] =  [10]
Determine the cost of a tray of club sandwiches and the cost of a tray of vegetarian sandwiches. SHOW YOUR WORK.

Answer 1

Answer: Cost of a tray of club sandwiches is $49 and Cost of a tray of vegetarian sandwiches is $70.

Explanation:

Since we have given that

`\begin{bmatrix} 7&10 \\ 0& 1\end{bmatrix}\begin{bmatrix}c \\ v\end{bmatrix}=\begin{bmatrix}149 \\ 10\end{bmatrix}`

since we know the formula for "System of linear of equation of matrix"

i.e.

`AX=B\\\\Here,\\\\A=\begin{bmatrix} 7&10 \\ 0& 1\end{bmatrix}\\\\X=\begin{bmatrix}c \\ v\end{bmatrix}\\\\B=\begin{bmatrix}149 \\ 10\end{bmatrix}`

Now, we will solve it :

`\begin{bmatrix}c \\\\ v\end{bmatrix}=\begin{bmatrix} 7&10 \\\\ 0& 1\end{bmatrix}^(-1)\begin{bmatrix}149 \\\\10\end{bmatrix}\\\\\begin{bmatrix}c \\\\ v\end{bmatrix}=\frac{1}{\det \begin{pmatrix}7&10\\ 0&1\end{pmatrix}}\begin{pmatrix}1&-10\\ -0&7\end{pmatrix}\begin{bmatrix}149 \\\\10\end{bmatrix}=(1)/(7)\begin{pmatrix}1&-10\\ -0&7\end{pmatrix}\begin{bmatrix}149 \\\\10\end{bmatrix}=\begin{bmatrix} 1&-10 \\\\ 0& 7\end{bmatrix}\begin{bmatrix}149 \\\\10\end{bmatrix}\\\\\begin{bmatrix}c \\\\v\end{bmatrix}=\begin{bmatrix}149-100\\\\70\end{bmatrix}\\\\\begin{bmatrix}c \\\\v\end{bmatrix}=\begin{bmatrix}49\\\\70\end{bmatrix}`

Hence Cost of a tray of club sandwiches is $49 and Cost of a tray of vegetarian sandwiches is $70.