Chapter 7.6, Problem 63E

To determine

To find : the numbers of bags of the three types of potting soil that the company can produce with given amount of raw materials

Answer to Problem 63E

The value of $x=100,y=100,z=100$

Explanation of Solution

Given information : The question is

  $500$ units of sand, $500$ units of loam and $400$ units of peat moss

Calculation :

Let,

  $x$ Is number of bags for seedling

  $y$ Is number of bags for general plotting

  $z$ Is number of bags for hardwood plant

Set up the equation

  $\begin{array}{l}2x+y+2z=500\leftarrow \text{ in terms of sand}\\ x+2y+2x=500\leftarrow \text{ in terms of loam}\\ x+y+2z=400\leftarrow \text{ in terms of peat moss}\\ AX=B\\ \left[\begin{array}{l}2\text{ 1 2}\\ 1\text{ 2 2}\\ 1\text{ 1 2}\end{array}\right]\left[\begin{array}{l}x\\ y\\ z\end{array}\right]=\left[\begin{array}{l}500\\ 500\\ 400\end{array}\right]\\ \left[A:I\right]=\left[\begin{array}{l}2\text{ 1 2 : 1 0 0 }\\ 1\text{ 2 2 : 0 1 0}\\ 1\text{ 1 2 : 0 0 1}\end{array}\right]\\ \left[A:I\right]\text{ to }\left[I:A^{-1}\right]\end{array}$

Using the elementary row operations

  $\begin{array}{l}-R_3\leftrightarrow R_2\\ R_3+3R_2\leftrightarrow R_3\\ \left[\begin{array}{l}1\text{ 2 2 : 0 1 0 }\\ 0\text{ 1 0 : 0 1 }-1\\ 0-3-\text{2 : 1 1 }-3\end{array}\right]\\ -\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.R_3\to R_3\&R_1-2R_3\to R_1\\ \left[\begin{array}{l}1\text{ 2 0 : 1 2 }-3\\ 0\text{ 1 0 : 0 1 }-1\\ 001\text{ : }-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.-\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\end{array}\right]\\ R_1-2R_2\to R_1\\ \left[\begin{array}{l}1\text{ 0 0 : 1 0 }-1\\ 0\text{ 1 0 : 0 1 }-1\\ 001\text{ : }-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\end{array}\right]\\ X=A^{-1}B\\ X=\left[\begin{array}{l}1\text{ 0 }-1\\ 0\text{ 1 }-1\\ -\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\end{array}\right]\left[\begin{array}{l}500\\ 500\\ 400\end{array}\right]\\ X=\left[\begin{array}{l}100\\ 100\\ 100\end{array}\right]\to x=100,y=100,z=100\end{array}$