Chapter 9.9, Problem 21WE

To determine

To calculate: To show the given system is inconsistent

Answer to Problem 21WE

The given system is inconsistent.

Explanation of Solution

Given:

  $\begin{array}{l}x-2y-z=-1\\ 2x-y+z=3\\ x+4y+5z=5\end{array}$

Calculation:

Given equation are as follows:

  $\begin{array}{l}x-2y-z=-\mathrm{1.....................}(1)\\ 2x-y+z=\mathrm{3.....................}(2)\\ x+4y+5z=\mathrm{5.....................}(3)\end{array}$

Add equation (1) and (2):

  $\begin{array}{l}x-2y-z=-\mathrm{1.....................}from(1)\\ 2x-y+z=\mathrm{3.......................}from(2)\\ \_\_\_\_\_\_\_\_\_\_\_\_\_\\ 3x-3y=2\\ x-y=\frac{2}{3}\mathrm{.....................}(4)\end{array}$

Multiply equation (1) by 5 and add it to equation (3):

  $\begin{array}{l}5x-10y-5z=-\mathrm{5.....................}Equation(1)\times 5\\ x+4y+5z=\mathrm{5.....................}from(3)\\ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\\ 6x-6y=0\\ 6(x-y)=0\\ x-y=\mathrm{0.....................}(5)\\ \end{array}$

Subtract equation (4) from (5):

  $\begin{array}{l}x-y=\frac{2}{3}\\ x-y=0\\ \_\_\_\_\_\_\_\_\\ 0=\frac{2}{3}\mathrm{..............}(6)\end{array}$

Equation (6) is a contradiction. Hence the system is inconsistent.

Conclusion:

The given system is inconsistent.