In bookstore A, a fiction book costs $8, a non-fiction book costs$10 and a science book costs $11.50. In bookstore B, a fiction book costs $1.10 less, a non-fiction book costs$2.80 less and a science book costs $1.50 more than that in bookstore A. The information can be represented by the matrix

1. Form the purchase matrix with entries in the order Fiction, Non-fiction, Science for each buyer.

Kevin bought $4$ fiction, $3$ non-fiction and $2$ science books, so his row is $\begin{pmatrix}4&3&2\end{pmatrix}$.

Molly bought $2$ fiction books and $x$ science books. Since no non-fiction books are stated, her row is $\begin{pmatrix}2&0&x\end{pmatrix}$.

Hence
$P=\begin{pmatrix}4&3&2\\2&0&x\end{pmatrix}.$

1. Write the price matrix $Q$ in the correct order of book types and bookstores.

From the question,
$Q=\begin{pmatrix}8&6.90\\10&7.20\\11.50&13.00\end{pmatrix}$
where the rows are Fiction, Non-fiction and Science, and the columns are bookstores A and B.

2. Multiply $P$ and $Q$ to find each element of $R$.

Using
$P=\begin{pmatrix}4&3&2\\2&0&x\end{pmatrix},$
$R=PQ=\begin{pmatrix}4&3&2\\2&0&x\end{pmatrix}\begin{pmatrix}8&6.90\\10&7.20\\11.50&13.00\end{pmatrix}$
$=\begin{pmatrix}4(8)+3(10)+2(11.50)&4(6.90)+3(7.20)+2(13.00)\\2(8)+0(10)+x(11.50)&2(6.90)+0(7.20)+x(13.00)\end{pmatrix}$

3. Simplify the numerical and algebraic entries.

$R=\begin{pmatrix}85&75.20\\16+11.5x&13.80+13x\end{pmatrix}$

1. Identify what the first column of $R$ corresponds to.

In $R=PQ$, the columns of $R$ correspond to the bookstores. Hence the first column corresponds to bookstore A.

2. State the contextual meaning of the entries in the first column.

The first column entries are $85$ and $16+11.5x$, so they represent the total costs of Kevin's and Molly's purchases respectively in bookstore A.

1. Form and solve the equation comparing Molly's total costs in bookstores B and A.

From part (b), Molly's total cost in bookstore A is $16+11.5x$ and in bookstore B is $13.80+13x$.

Since she saves $5.30 by purchasing in bookstore A,