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

Answer:

Let us assume cost of each pineapple cake = $x.

Cost of each chocolate cakes = $y.

On weekdays 45 pineapple cakes and 62 chocolate cakes were sold.

Total cost of 45 pineapple cakes and 62 chocolate cakes were sold =  $3,520.

Therefore, first equation would be

45x + 62y = 3520  --------------equation(1).

On weekends 55 pineapple cakes and 79 chocolate cakes were sold.

Total cost of 55 pineapple cakes and 79 chocolate cakes were sold =  $4,415.

Therefore, second equation would be

55x + 79y = 4415   --------------equation(2).

Therefore, system of equation are 45x + 62y = 3520  and 55x + 79y = 4415.

And we can form a matrix as

.

Step-by-step explanation:

Answer 2

Answer:

• 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

Step-by-step 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:

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

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

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:

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

and the price of the pineapple cake is given by:

$x=\dfrac{|A_x|}{|A|}$

$x=\dfrac{4350}{145}\\\\\\x=30$

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:

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

and the price of one chocolate cake is given by:

$y=\dfrac{|A_y|}{|A|}$

$y=\dfrac{5075}{145}\\\\\\y=35$