Chapter 3.4, Problem 30STP

To determine

To choose: The correct option by solving the given system of equations

  $\begin{array}{l}x-y+z=\mathrm{0....}(1)\\ -5x+3y-2z=-\mathrm{1...........}(2)\\ 2x-y+4z=\mathrm{11...........}(3)\end{array}$

Answer to Problem 30STP

On solving it is known that option B is correct.

Explanation of Solution

Given information :

The system of equation;

  $\begin{array}{l}x-y+z=\mathrm{0....}(1)\\ -5x+3y-2z=-\mathrm{1...........}(2)\\ 2x-y+4z=\mathrm{11...........}(3)\end{array}$

Calculation :

Consider the system of equation;

  $\begin{array}{l}x-y+z=\mathrm{0....}(1)\\ -5x+3y-2z=-\mathrm{1...........}(2)\\ 2x-y+4z=\mathrm{11...........}(3)\end{array}$

The system of equation is solved as follows:

Multiple with 3 equation (1) and add with (2) as follows:

  $\begin{array}{r}\hfill 3x-3y+3z=0\\ \hfill -5x+3y-2z=-1\\ \hfill -2x+z=-1\end{array}$

Multiple with 3 equations (3) and add with (2) as follows:

  $\begin{array}{r}\hfill 6x-3y+12z=33\\ \hfill -5x+3y-2z=-1\\ \hfill x+10z=32\end{array}$

Multiple with 2 equation $(2)-2x+z=-1$ and add with $x+10z=32$ as follows:

  $\begin{array}{r}\hfill -2x+z=-1\\ \hfill 2x+20z=64\\ \hfill 21z=63\\ \hfill z=3\end{array}$

Substitute $z=3$ in equation $-2x+z=-1$ as follows:

  $\begin{array}{l}-2x+3=-1\\ \begin{array}{cc}-2x& =-4\\ x& =2\end{array}\end{array}$

Substitute $x=2$ and $z=3$ in equation (1):

  $\begin{array}{l}2-y+3=0\\ \begin{array}{c}-y=-5\\ y=5\end{array}\end{array}$

Therefore, the solution set is $\left(2,5,3\right).$

Hence option B is correct.