Question

can anyone help me??? A bakery sells muffins for $3.50 each. A beverage is $1.75. A class purchases 32 items and spends a total of $87.50. Write the system of equations and represent it as a matrix equation. b. State the value of the determinant. c. Use matrices to solve the system. Find the number of muffins and the number of beverages purchased.

Answer 1

Not sure about matrix equation but this is how I answer it.

Given:
muffins = 3.50
beverage = 1.75
total number of items = 32
total cost = 87.50

m + b = 32
3.50m + 1.75b = 87.50

m = 32 - b

3.50(32-b) + 1.75b = 87.50
112 - 3.50b + 1.75b = 87.50
-3.50b + 1.75b = 87.50 - 112
-1.75b = -24.50
b = -24.50 / -1.75
b = 14

m = 32 - b
m = 18

The class bought 18 muffins and 14 beverages.

3.50m + 1.75b = 87.50
3.50(18) + 1.75(14) = 87.50
63 + 24.50 = 87.50
87.50 = 87.50

Answer 2

(a)
`\begin{bmatrix} 1 &1 \\ 2 & 1\end{bmatrix}\cdot\begin{bmatrix}x\\ y\end{bmatrix}=\begin{bmatrix}32\\ 50\end{bmatrix}`

(b) determinant of the matrix is: -1

(c) number of muffins purchased=18

number of beverages purchased=14.

Explanation:

(a) let us assume that the number of muffins purchased=x

number of beverage purchased=y

as a class purchased a total of 32 items that means x+y=32

amount spent=$87.50 that means 3.50x+1.75y=87.50

which could also be written as: 2x+y=50

now in matrix form it could be represented as:

`\begin{bmatrix} 1 &1 \\ 2 & 1\end{bmatrix}\cdot\begin{bmatrix}x\\ y\end{bmatrix}=\begin{bmatrix}32\\ 50\end{bmatrix}`

(b) the determinant of the matrix
`\begin{bmatrix} 1 &1 \\ 2 & 1\end{bmatrix}` is given by
`1*1-2*1=1-2=-1`

hence the determinant of the matrix is: -1

(c) now on solving the matrix we need to apply row operations in it to make it a upper triangular matrix so that it would be easy for us to calculate the value of x and y.

`\begin{bmatrix} 1 &1 \\ 2 & 1\end{bmatrix}\cdot\begin{bmatrix}x\\ y\end{bmatrix}=\begin{bmatrix}32\\ 50\end{bmatrix}`

we multiply row 1 by 2 and subtract row 2 from it to get

`\begin{bmatrix} 1 &1 \\ 0 & 1\end{bmatrix}\cdot\begin{bmatrix}x\\ y\end{bmatrix}=\begin{bmatrix}32\\ 14\end{bmatrix}`

on using the backward substitution in the matrix we get y=14

and x=18

Hence, number of muffins purchased=18

number of beverages purchased=14.