Chapter 4, Problem 34SGR

To determine

To find: The cost of 1 pound of nuts and 1 pounds of raisins.

Answer to Problem 34SGR

The cost of one pound of nuts is $\$5.25$ and one pound of raisins is $\$6.5$ .

Explanation of Solution

Given:

The cost of two pounds of nuts and two pounds of raisins is $\$23.50$ .

The cost of three pounds of nuts and 1 pound of raisins is $\$22.5$ .

Calculation:

Consider the cost of one pound of nuts is $x$ and cost of one pound of raisins is $y$ .

Consider the matrix for the given relation is,

  $\left[\begin{array}{cc}2& 2\\ 3& 1\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}23.50\\ 22.25\end{array}\right]$

Solve for the inverse of the coefficient matrix as,

  $2\left(1\right)-2\left(3\right)=-4$

Since, the determinant is not zero, it is solved as the inverse of the coefficient matrix as.,

  $\begin{array}{c}-\frac{1}{4}\left[\begin{array}{cc}1& -2\\ -3& 2\end{array}\right]=\left[\begin{array}{cc}\left(-\frac{1}{4}\right)1& -2\left(-\frac{1}{4}\right)\\ -3\left(-\frac{1}{4}\right)& 2\left(-\frac{1}{4}\right)\end{array}\right]\\ =\left[\begin{array}{cc}-\frac{1}{4}& \frac{2}{4}\\ \frac{3}{4}& -\frac{2}{4}\end{array}\right]\end{array}$

Then,

  $\begin{array}{l}\left[\begin{array}{cc}-\frac{1}{4}& \frac{2}{4}\\ \frac{3}{4}& -\frac{2}{4}\end{array}\right]\left[\begin{array}{cc}2& 2\\ 3& 1\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{cc}-\frac{1}{4}& \frac{2}{4}\\ \frac{3}{4}& -\frac{2}{4}\end{array}\right]\left[\begin{array}{c}23.50\\ 22.25\end{array}\right]\\ \left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}-\frac{1}{4}\left(23.50\right)+\frac{2}{4}\left(22.25\right)\\ \frac{3}{4}\left(23.50\right)-\frac{2}{4}\left(22.25\right)\end{array}\right]\\ \left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}\$5.25\\ \$6.5\end{array}\right]\end{array}$