Chapter 9.9, Problem 7ST

To determine

To Solve −

  $2x-y+z=11$

  $x+2y+3z=8$

  $3x-4y-5z=-2$

Answer to Problem 7ST

  $x=2$ , $y=-3$ and $z=4$ .

Explanation of Solution

Given:

  $2x-y+z=11$

  $x+2y+3z=8$

  $3x-4y-5z=-2$

Calculation:

Given:

  $2x-y+z=11$

  $x+2y+3z=8$

  $3x-4y-5z=-2$

Multiplying equation $x+2y+3z=8$ by $2$ , we have:

  $2x+4y+6z=16$

Solving the equation $2x-y+z=11$ and $2x+4y+6z=16$ , we have:

  $2x=16-4y-6z=11+y-z$

Solving further, we have:

  $\begin{array}{l}16-4y-6z=11+y-z\\ -5y-5z=-5\\ y+z=1\end{array}$

Multiplying equation $x+2y+3z=8$ by $3$ , we have:

  $3x+6y+9z=24$

Solving the equation $3x-4y-5z=-2$ and $3x+6y+9z=24$ , we have:

  $3x=24-6y-9z=-2+4y+5z$

Solving further, we have:

  $\begin{array}{l}24-6y-9z=-2+4y+5z\\ -10y-14z=-26\\ 5y+7z=13\end{array}$

Solving the equation $y+z=1$ and $5y+7z=13$ , we have:

  $\begin{array}{l}5\left(1-z\right)+7z=13\\ 5-5z+7z-13=0\\ 2z=8\\ z=4\end{array}$

Solving the equation $y+z=1$ for value of $y$ , we have:

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

Solving the equation $x+2y+3z=8$ for value of $x$ , we have:

  $\begin{array}{l}x+2y+3z=8\\ x-6+12=8\\ x=2\end{array}$

Thus, $x=2$ , $y=-3$ and $z=4$ .

Conclusion:

  $x=2$ , $y=-3$ and $z=4$ .