Chapter 2.5, Problem 51E

To determine

To find: the cost complete computer system and printers.

Answer to Problem 51E

Cost of computer = $959

Cost of printer = $239

Explanation of Solution

Given information:

Let x and y be the cost of computer and printer respectively.

The equations depicting the data for total sale on first second and third day are as follows:

  $38x+53y=49109$

  $22x+55y=31614$

  $21x+26y=26353$

Calculation:

Set up the matrix for the first two equations as follows:

  $\begin{array}{l}\begin{array}{ccc}A& X& B\end{array}\\ \left[\begin{array}{cc}\begin{array}{l}38\\ 22\end{array}& \begin{array}{l}53\\ 44\end{array}\end{array}\right]\left[\begin{array}{l}x\\ y\end{array}\right]=\left[\begin{array}{l}49109\\ 31614\end{array}\right]\end{array}$

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:

  $\begin{array}{c}{A}^{-1}=\frac{1}{1672-1166}\left[\begin{array}{cc}\begin{array}{l}44\\ -53\end{array}& \begin{array}{l}-22\\ \text{ 38}\end{array}\end{array}\right]\\ \\ =\frac{1}{506}\left[\begin{array}{cc}\begin{array}{l}44\\ -53\end{array}& \begin{array}{l}-22\\ \text{ 38}\end{array}\end{array}\right]\end{array}$

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

  $\begin{array}{c}\frac{1}{506}\left[\begin{array}{cc}\begin{array}{l}44\\ -53\end{array}& \begin{array}{l}-22\\ \text{ 38}\end{array}\end{array}\right]\cdot \left[\begin{array}{l}49109\\ 31614\end{array}\right]=\left[\begin{array}{l}x\\ y\end{array}\right]\\ \left[\begin{array}{l}959\\ 259\end{array}\right]=\left[\begin{array}{l}x\\ y\end{array}\right]\end{array}$

Thus, the value of x and y is $959 and $259 respectively.