Chapter 10.5, Problem 49E

To determine

To find: The system of equation according to given information in term of $x,y\text{ and z}\text{.}$

Answer to Problem 49E

  $\begin{array}{l}x+y+2z=675\\ 2x+y+z=600\\ x+2y+z=625\end{array}$

Explanation of Solution

Given: An encyclopedia saleswoman works for a company that offers three different grades of bindings for its encyclopedias: standard, deluxe and leather. For each set she sells, she earns a commission based on the set’s binding grade. One week she sells one standard, one deluxe and two leather sets and makes $675 in commission. The next week she sells two standard, one deluxe and one leather set for a $600 commission. The third week she sells one standard , two deluxe and one leather earning $625 in commission.

Let $x,y\text{ and }z$ represent the commission she earns on standard, deluxe and leather sets respectively.

According to question,

First week she sells one standard, one deluxe and two leather sets and makes $675 in commission.

  $x+y+2z=675$

The next week she sells two standard, one deluxe and one leather set for a $600 commission.

  $2x+y+z=600$

The third week she sells one standard , two deluxe and one leather earning $625 in commission.

  $x+2y+z=625$

Hence, the system of equations are

  $\begin{array}{l}x+y+2z=675\\ 2x+y+z=600\\ x+2y+z=625\end{array}$

To determine

To express: The system of equations in matrix form $AX=B$ .

Answer to Problem 49E

  $\left[\begin{array}{ccc}1& 1& 2\\ 2& 1& 1\\ 1& 2& 1\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}675\\ 600\\ 625\end{array}\right]$

Explanation of Solution

Given: The system of equations are,

  $\begin{array}{l}x+y+2z=675\\ 2x+y+z=600\\ x+2y+z=625\end{array}$

The matrix form $AX=B$

A is matrix form by coefficient of variable $x,y\text{ and z}\text{.}$

  $A=\left[\begin{array}{ccc}1& 1& 2\\ 2& 1& 1\\ 1& 2& 1\end{array}\right]$

X is matrix of variable.

  $X=\left[\begin{array}{c}x\\ y\\ z\end{array}\right]$

B is matrix of constant term.

  $B=\left[\begin{array}{c}675\\ 600\\ 625\end{array}\right]$

Now using above matrix to write system of equation in matrix form.

  $\left[\begin{array}{ccc}1& 1& 2\\ 2& 1& 1\\ 1& 2& 1\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}675\\ 600\\ 625\end{array}\right]$

To determine

To find: The commission of each type of binding.

Answer to Problem 49E

Commission on standard type of encyclopedias binding is $125

Commission on deluxe type of encyclopedias binding is $150

Commission on leather type of encyclopedias binding is $200

Explanation of Solution

Given: The system of equations are,

  $\begin{array}{l}x+y+2z=675\\ 2x+y+z=600\\ x+2y+z=625\end{array}$

  $A=\left[\begin{array}{ccc}1& 1& 2\\ 2& 1& 1\\ 1& 2& 1\end{array}\right]$

First find the inverse of this matrix using row-echelon method.

  $\begin{array}{l}\left[\begin{array}{ccc}1& 1& 2\\ 2& 1& 1\\ 1& 2& 1\end{array}\right]A^{-1}=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\\ R_2\to R_2-2R_1,R_3\to R_3-R_1\\ \left[\begin{array}{ccc}1& 1& 2\\ 0& -1& -3\\ 0& 1& -1\end{array}\right]A^{-1}=\left[\begin{array}{ccc}1& 0& 0\\ -2& 1& 0\\ -1& 0& 1\end{array}\right]\end{array}$

Therefore, $A^{-1}=\left[\begin{array}{ccc}-\frac{1}{4}& \frac{3}{4}& -\frac{1}{4}\\ -\frac{1}{4}& -\frac{1}{4}& \frac{3}{4}\\ \frac{3}{4}& -\frac{1}{4}& -\frac{1}{4}\end{array}\right]$

The matrix form of system of equation, $AX=B$

  $\begin{array}{l}X=A^{-1}B\\ \left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{ccc}-\frac{1}{4}& \frac{3}{4}& -\frac{1}{4}\\ -\frac{1}{4}& -\frac{1}{4}& \frac{3}{4}\\ \frac{3}{4}& -\frac{1}{4}& -\frac{1}{4}\end{array}\right]\left[\begin{array}{c}675\\ 600\\ 625\end{array}\right]\\ \left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}125\\ 150\\ 200\end{array}\right]\\ x=125,y=150,z=200\end{array}$

Hence, commission on standard type of encyclopedias binding is $125

Commission on deluxe type of encyclopedias binding is $150

Commission on leather type of encyclopedias binding is $200