Chapter 9.5, Problem 44WE

To determine

To find: The value of $x,y\text{ and }z$ by using multiplication with the addition-or-subtraction method.

Answer to Problem 44WE

The the value of $x$ is $1$ , the value of $y$ is $2$ , and the value of $z$ is $4$ .

Explanation of Solution

Given information:

The given equations are,

  $\begin{array}{c}x+y+z=7\\ 2x-y+z=4\\ 3x+2y+z=11\end{array}$

Calculation:

Calculate the equations,

  $\begin{array}{c}x+y+z=\mathrm{7................................}\left(1\right)\\ 2x-y+z=\mathrm{4................................}\left(2\right)\\ 3x+2y+z=\mathrm{11..............................}\left(3\right)\end{array}$

Add equation $\left(1\right)\&\left(2\right)$ .

  $\begin{array}{c}x+y+z=7\\ 2x-y+z=4\end{array}$

  $3x+2z\text{ = 11}$

Solve the result,

  $3x+2z=\mathrm{11..............................}\left(4\right)$

Multiply by $2$ in equation $\left(1\right)$

  $\begin{array}{c}2\left(x+y+z\right)=2\left(7\right)\\ 2x+2y+2z=\mathrm{14.............................}\left(5\right)\end{array}$

Subtract equation $\left(5\right)\&\left(3\right)$ .

  $\begin{array}{r}2x+2y+2z=14\\ 3x+2y+z=11\\ ----\end{array}$

  $-x+z\text{ = 3}$

Solve the result,

  $-x+z=\mathrm{3..............................}\left(6\right)$

Multiply by $3$ in equation $\left(6\right)$ .

  $\begin{array}{c}3\left(-x+z\right)=3\left(3\right)\\ -3x+3z=\mathrm{9......................}\left(7\right)\end{array}$

Add equation $\left(4\right)$ and equation $\left(7\right)$ .

  $\begin{array}{c}\text{3}x+2z=11\\ -3x+3z=9\end{array}$

  $5z\text{ =20}$

Solve the result,

  $z=4$

Substitute the value of $z$ in equation $\left(4\right)or\left(7\right)$ to get $z$ .

  $\begin{array}{c}3x+2z=11\\ 3x+2\left(4\right)=11\\ 3x+8=11\\ 3x=3\\ x=1\end{array}$

Substitute the value of $x\&z$ in original equation get $y$ .

  $\begin{array}{c}x+y+z=7\\ 1+y+4=7\\ y=7-5\\ y=2\end{array}$

Therefore,

  $\begin{array}{l}x=1\\ y=2\\ z=4\end{array}$

Therefore, the value of $x$ is $1$ , the value of $y$ is $2$ , and the value of $z$ is $4$ .