Chapter 4.6, Problem 3CYP

To determine

To find: the price in dollars of a new comic book.

Answer to Problem 3CYP

The price is $\$2.875$ .

Explanation of Solution

Given information:

Money spent by Dante and Erica is $\$11.25$ and $\$15.75$ respectively.

Dante bought 3 new and 4 old books whereas Erica bought 10 old and 3 new books.

Formula used:

Inverse of a $2\times 2$ matrix $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ is $A^{-1}=\frac{1}{ad-bc}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$ is, where $ad-bc\ne 0$ .

The value of a second order determinant is the difference of the products of the products of the two diagonals.

Steps to solve the equations are:-

Step1. Find the inverse of the coefficient matrix.

Step 2. Multiple each side of the matrix equation by the inverse matrix.

Calculation:

According to the question ,

	Money spent	No. of old books	NO. of new books
Dante	$\$11.25$	4	3
Erica	$\$15.75$	10	3

Let x be the number of old books bought.

And y be the number of new books bought.

Thus, the system of equation formed is

  $\begin{array}{l}4x+3y=11.25\\ 10x+3y=15.75\end{array}$

The matrix equation is

  $\left[\begin{array}{cc}4& 3\\ 10& 3\end{array}\right].\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}11.25\\ 15.75\end{array}\right]$ .

Step 1.

  $=A^{-1}=\frac{1}{(4)(3)-(3)(10)}\left[\begin{array}{cc}3& -3\\ -10& 4\end{array}\right]$

  $\begin{array}{l}=\frac{1}{12-30}\left[\begin{array}{cc}3& -3\\ -10& 4\end{array}\right]\\ =-\frac{1}{18}\left[\begin{array}{cc}3& -3\\ -10& 4\end{array}\right]\end{array}$

Step2.

  $\begin{array}{l}=-\frac{1}{18}\left[\begin{array}{cc}3& -3\\ -10& 4\end{array}\right]\left[\begin{array}{cc}4& 3\\ 10& 3\end{array}\right].\left[\begin{array}{c}x\\ y\end{array}\right]=-\frac{1}{18}\left[\begin{array}{cc}3& -3\\ -10& 4\end{array}\right]\left[\begin{array}{c}11.25\\ 15.75\end{array}\right]\\ =\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=-\frac{1}{18}\left[\begin{array}{c}(3\times 11.25)+(-3\times 15.75)\\ (-10\times 11.25)+(4\times 15.75)\end{array}\right]\\ =\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=-\frac{1}{18}\left[\begin{array}{c}33.75-47.25\\ -112.5+63\end{array}\right]\\ =\left[\begin{array}{c}x\\ y\end{array}\right]=-\frac{1}{18}\left[\begin{array}{c}-13.5\\ -51.75\end{array}\right]\end{array}$

Or

  $=\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}0.75\\ 2.875\end{array}\right]$

Thus, the price of new book is $\$2.875$ .