Chapter 2.5, Problem 46E

To determine

To find: the cars made in year 1994 and 1995.

Answer to Problem 46E

  $240000\text{ cars in 1994 and 15}0000\text{ cars in 1995}$

Explanation of Solution

Given information:

Let x and y be the number of cars produced in year 1994 and 1995.

The equations depicting the data are as follows:

  $\begin{array}{l}x+y=390000\\ x=90000+y\to x-y=90000\end{array}$

Calculation:

Set up the matrix for the given equations as follows:

  $\begin{array}{l}\begin{array}{ccc}A& X& B\end{array}\\ \left[\begin{array}{cc}\begin{array}{l}1\\ 1\end{array}& \begin{array}{l}1\\ -1\end{array}\end{array}\right]\left[\begin{array}{l}x\\ y\end{array}\right]=\left[\begin{array}{l}390000\\ 90000\end{array}\right]\end{array}$

For $A=\left[\begin{array}{cc}\begin{array}{l}a\\ c\end{array}& \begin{array}{l}b\\ d\end{array}\end{array}\right]$ , the inverse formula is $A^{-1}=\frac{1}{ad-bc}\left[\begin{array}{cc}\begin{array}{l}d\\ -c\end{array}& \begin{array}{l}-b\\ a\end{array}\end{array}\right]$ . Put the values in the above formula and evaluate the inverse for the given matrix as follows:

  $\begin{array}{c}{A}^{-1}=\frac{1}{-1-1}\left[\begin{array}{cc}\begin{array}{l}-1\\ -1\end{array}& \begin{array}{l}-1\\ 1\end{array}\end{array}\right]\\ =-\frac{1}{2}\left[\begin{array}{cc}\begin{array}{l}-1\\ -1\end{array}& \begin{array}{l}-1\\ 1\end{array}\end{array}\right]\end{array}$

Solve $A^{-1}B=X$ as follows:

  $\begin{array}{c}-\frac{1}{2}\left[\begin{array}{cc}\begin{array}{l}-1\\ -1\end{array}& \begin{array}{l}-1\\ 1\end{array}\end{array}\right]\cdot \left[\begin{array}{l}390000\\ 90000\end{array}\right]=\left[\begin{array}{l}x\\ y\end{array}\right]\\ \left[\begin{array}{l}240000\\ 150000\end{array}\right]=\left[\begin{array}{l}x\\ y\end{array}\right]\end{array}$

Thus, the value of x and y is 240000 cars and 150000 respectively.