Respuesta :
Answer:
- The determinant of the matrix for the pineapple cake is:
[tex]\left[\begin{array}{ccc}3520&62\\4415&79\end{array}\right][/tex]
- The price of one pineapple cake is $ 30
- The determinant of the matrix for the chocolate cake is:
[tex]\left[\begin{array}{ccc}45&3520\\55&4415\end{array}\right][/tex]
- 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:
[tex]Ax=b[/tex]
where
[tex]A=\left[\begin{array}{ccc}45&62\\55&79\end{array}\right][/tex]
Determinant of A is:
[tex]|A|=\begin{vmatrix}45&62\\55&79\end{vmatrix}\\\\i.e.\\\\|A|=45\times 79-55\times 62\\\\\\|A|=145[/tex]
[tex]x=\left[\begin{array}{ccc}x\\y\end{array}\right][/tex]
and [tex]b=\left[\begin{array}{ccc}3520\\4415\end{array}\right][/tex]
Hence, the matrix of the pineapple cake is given by:
[tex]A_x=\left[\begin{array}{ccc}3520&62\\4415&79\end{array}\right][/tex]
and its determinant is given by:
[tex]|A_x|=\begin{vmatrix}3520&62\\4415&79\end{vmatrix}\\ \\|A_x|=3520\times 79-4415\times 62\\\\|A_x|=4350[/tex]
and the price of the pineapple cake is given by:
[tex]x=\dfrac{|A_x|}{|A|}[/tex]
[tex]x=\dfrac{4350}{145}\\\\\\x=30[/tex]
The matrix of the chocolate cake is given by:
[tex]A_y=\left[\begin{array}{ccc}45&3520\\55&4415\end{array}\right][/tex]
and its determinant is given by:
[tex]|A_y|=\begin{vmatrix}45&3520\\55&4415\end{vmatrix}\\ \\|A_y|=4415\times 45-3520\times 55\\\\|A_y|=5075[/tex]
and the price of one chocolate cake is given by:
[tex]y=\dfrac{|A_y|}{|A|}[/tex]
[tex]y=\dfrac{5075}{145}\\\\\\y=35[/tex]
