Question

One week, a cake shop sold 45 pineapple cakes and 62 chocolate cakes on the weekdays and 55 pineapple cakes and 79 chocolate cakes over the weekend. Total sales were $3,520 on the weekdays and $4,415 over the weekend. Form a linear system representing this data and create a matrix. The determinant of the matrix for the pineapple cake is . The price of one pineapple cake is $. The determinant of the matrix for the chocolate cake is . The price of one chocolate cake is $.

Answer 1

• The determinant of the matrix for the pineapple cake is:

`\left[\begin{array}{ccc}3520&62\\4415&79\end{array}\right]`

• The price of one pineapple cake is $ 30

• The determinant of the matrix for the chocolate cake is:

`\left[\begin{array}{ccc}45&3520\\55&4415\end{array}\right]`

• The price of one chocolate cake is $ 35

Explanation:

Let the price of one pineapple cake is: $ x

and the price of one chocolate cake is: $ y

Now, 45 pineapple cakes and 62 chocolate cakes on the weekdays.

such that total sales were $3,520 on the weekdays.

This means the equation that can be formed is:

45x+62y=3520

Also, 55 pineapple cakes and 79 chocolate cakes over the weekend.

Total sales on weekend were $ 4415

This means that the equation is given by:

55x+79y=4415

Hence, we can form a system as:

`Ax=b`

where

`A=\left[\begin{array}{ccc}45&62\\55&79\end{array}\right]`

Determinant of A is:

`|A|=\begin{vmatrix}45&62\\55&79\end{vmatrix}\\\\i.e.\\\\|A|=45* 79-55* 62\\\\\\|A|=145`

`x=\left[\begin{array}{ccc}x\\y\end{array}\right]`

and
`b=\left[\begin{array}{ccc}3520\\4415\end{array}\right]`

Hence, the matrix of the pineapple cake is given by:

`A_x=\left[\begin{array}{ccc}3520&62\\4415&79\end{array}\right]`

and its determinant is given by:

`|A_x|=\begin{vmatrix}3520&62\\4415&79\end{vmatrix}\\ \\|A_x|=3520* 79-4415* 62\\\\|A_x|=4350`

and the price of the pineapple cake is given by:

`x=(|A_x|)/(|A|)`

`x=(4350)/(145)\\\\\\x=30`

The matrix of the chocolate cake is given by:

`A_y=\left[\begin{array}{ccc}45&3520\\55&4415\end{array}\right]`

and its determinant is given by:

`|A_y|=\begin{vmatrix}45&3520\\55&4415\end{vmatrix}\\ \\|A_y|=4415* 45-3520* 55\\\\|A_y|=5075`

and the price of one chocolate cake is given by:

`y=(|A_y|)/(|A|)`

`y=(5075)/(145)\\\\\\y=35`