Chapter 9.9, Problem 22WE

To determine

To calculate: To show the given system is inconsistent

Answer to Problem 22WE

The given system is inconsistent.

Explanation of Solution

Given:

  $\begin{array}{l}x+2y-2z=3\\ x+3y-4z=6\\ 4x+5y-2z=6\end{array}$

Calculation:

Given equation are as follows:

  $\begin{array}{l}x+2y-2z=\mathrm{3.....................}(1)\\ x+3y-4z=\mathrm{6.....................}(2)\\ 4x+5y-2z=\mathrm{6.....................}(3)\end{array}$

Subtract equation (1) from (2):

  $\begin{array}{l}x+3y-4z=\mathrm{6.....................}from(2)\\ x+2y-2z=\mathrm{3.....................}from(1)\\ \_\_\_\_\_\_\_\_\_\_\_\_\_\\ y-2z=\mathrm{3.....................}(4)\end{array}$

Multiply equation (1) by 4 and subtract equation (3) from it:

  $\begin{array}{l}4x+8y-8z=\mathrm{12.....................}Equation(1)\times 4\\ 4x+5y-2z=\mathrm{6.....................}from(3)\\ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\\ 3y-6z=6\\ 3(y-2z)=6\\ y-2z=\mathrm{2....................}(5)\\ \end{array}$

Subtract equation (4) and (5):

  $\begin{array}{l}y-2z=3\\ y-2z=2\\ \_\_\_\_\_\_\_\_\\ 0=\mathrm{1..............}(6)\end{array}$

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

Conclusion:

The given system is inconsistent.