Given that $\displaystyle\begin{pmatrix}9 & 2\\[4pt]3 & 0\end{pmatrix}\begin{pmatrix}1\\[4pt]-2y\end{pmatrix}=\begin{pmatrix}x\\[4pt]2x-7\end{pmatrix}$, find the value of $x$ and $y$. Robert loves to eat Beef-burgers and Chicken-burgers.
The matrix, $T$, shows the total number of Beef-burgers and Chicken-burgers he consumed in a particular week. Beef     Chicken $T=\displaystyle\begin{pmatrix}5 & 3\\[4pt]1 & 2\end{pmatrix}$ Monday to Friday Saturday and Sunday A Beef-burger has 300 calories and a Chicken-burger has 270 calories. Represent these calories in a 2 x 1 column matrix C. Evaluate the matrix $N = TC$. State what the elements in $N$ represent. Evaluate the matrix $A=\frac{1}{7}\begin{pmatrix}1 & 1\end{pmatrix}N$. Answer $A=\begin{pmatrix} \;\;\;\;\end{pmatrix}$ Explain what the matrix $A$ represents. Answer ... ... Robert hopes to reduce his total calories intake by 50% on weekdays and maintain his total calories intake on weekend. Using matrix multiplication, calculate the total amount of calories he should consume in the following week. Answer ...

1. Multiply the matrix by the column vector.

$\begin{pmatrix}9&2\\3&0\end{pmatrix}\begin{pmatrix}1\\-2y\end{pmatrix}=\begin{pmatrix}9(1)+2(-2y)\\3(1)+0(-2y)\end{pmatrix}=\begin{pmatrix}9-4y\\3\end{pmatrix}$

2. Equate corresponding entries and solve for $x$ and $y$.

Since
$\begin{pmatrix}9-4y\\3\end{pmatrix}=\begin{pmatrix}x\\2x-7\end{pmatrix},$
compare the second entries:
$3=2x-7$
$2x=10$
$x=5$
Compare the first entries:
$9-4y=x$
$9-4y=5$
$-4y=-4$
$y=1$

1. Form the $2\times 1$ column matrix using the given calorie values in Beef–Chicken order.

Since the columns of $T$ are ordered as Beef, then Chicken,
$C=\begin{pmatrix}\text{calories in a Beef-burger}\\\text{calories in a Chicken-burger}\end{pmatrix}=\begin{pmatrix}300\\270\end{pmatrix}.$

1. Substitute the matrices $T$ and $C$ into $N=TC$.

From the question and part (b)(i),
$T=\begin{pmatrix}5&3\\1&2\end{pmatrix},\quad C=\begin{pmatrix}300\\270\end{pmatrix}$
So
$N=\begin{pmatrix}5&3\\1&2\end{pmatrix}\begin{pmatrix}300\\270\end{pmatrix}.$

2. Carry out row-by-column multiplication to find each entry of $N$.

$N=\begin{pmatrix}5(300)+3(270)\\1(300)+2(270)\end{pmatrix}$
$=\begin{pmatrix}1500+810\\300+540\end{pmatrix}$
$=\begin{pmatrix}2310\\840\end{pmatrix}.$

1. Interpret each entry of $N$ using the row categories of $T$.

Since
$N=TC=\begin{pmatrix}2310\\840\end{pmatrix},$
the first entry corresponds to the first row of $T$, which is Monday to Friday, and the second entry corresponds to the second row of $T$, which is Saturday and Sunday.

1. Substitute $N$ and evaluate the matrix product and scalar multiplication.

From part (b)(ii)(a),
$N=\begin{pmatrix}2310\\840\end{pmatrix}.$
$A=\frac{1}{7}\begin{pmatrix}1&1\end{pmatrix}\begin{pmatrix}2310\\840\end{pmatrix}$
$=\frac{1}{7}\begin{pmatrix}1(2310)+1(840)\end{pmatrix}$
$=\frac{1}{7}\begin{pmatrix}3150\end{pmatrix}$
$=\begin{pmatrix}450\end{pmatrix}$

1. Interpret the matrix operation used to obtain $A$.

Since $A=\frac{1}{7}\begin{pmatrix}1&1\end{pmatrix}N,$ the entries of $N$ are added to give the total calories for the whole week, then divided by $7$. Hence $A=\begin{pmatrix}450\end{pmatrix}$ represents the average number of calories consumed per day from the burgers in that week.

1. Form the matrix multiplication representing 50% of weekday calories and 100% of weekend calories.

From earlier, $N=\begin{pmatrix}2310\\840\end{pmatrix}.$ Since weekday intake is reduced by $50\%$, Robert consumes $50\%=0.5$ of the weekday calories. Weekend intake is maintained, so he consumes $100\%=1$ of the weekend calories.

$\text{Total}=\begin{pmatrix}0.5&1\end{pmatrix}\begin{pmatrix}2310\\840\end{pmatrix}$

2. Evaluate the matrix product to obtain the total calorie intake.

$\begin{pmatrix}0.5&1\end{pmatrix}\begin{pmatrix}2310\\840\end{pmatrix}=0.5(2310)+1(840)$
$=1155+840=1995$