Chapter 7.1, Problem 67E

To determine

To solve: the system of equations.

Answer to Problem 67E

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

Explanation of Solution

Given information:

  $x+y-2z=3$............................ (1)

  $-4x-y-z=0$............................ (2)

  $-x-5y+4z=11$............................ (3)

Calculation:

Multiply equation (2) by $-2$ and then add the resultant equation to equation (1).

  $\begin{array}{l}\left(-4x-y-z=0\right)\left(-2\right)\to -8x+2y+2z=0\\ \\ x+y-2z=3\\ \underset{\_}{-8x+2y+2z=0}\\ 9x+3y=3\mathrm{...}\left(4\right)\end{array}$

Multiply equation (2) by 4 and then add the resultant equation to equation (3).

  $\begin{array}{l}\left(-4x-y-z=0\right)\left(4\right)\to -16x-4y-4z=0\\ \\ -x-5y+4z=11\\ \underset{\_}{-16x-4y-4z=0}\\ -17x-9y=11\mathrm{...}\left(5\right)\end{array}$

Now multiply equation (4) by 3 and then add it to equation (5).

  $\begin{array}{l}\left(9x+3y=3\right)\left(3\right)\to 27x+9y=9\\ \\ -17x-9y=11\\ \underset{\_}{27x+9y=9}\\ \text{ 10}x+0=20\\ x=2\left[\text{Divide both sides by 10}\right]\end{array}$

Put the value of x in equation (4) and solve y .

  $\begin{array}{c}9\left(2\right)+3y=3\\ 18+3y=3\\ 6+y=1\left[\text{Divide each side by 3}\right]\\ y=1-6\\ =-5\end{array}$

Put the value of $x\text{ and }y$ in equation (1) and evaluate z as shown below.

  $\begin{array}{c}-4x-y-z=0\\ z=-4x-y\left[\text{Rearranged the equation}\right]\\ =-4\left(2\right)-\left(-5\right)\left[\text{Substitute }x\text{ and }y\right]\\ =-8+5\\ =-3\end{array}$

Thus the value of $x,y$ and $z$ is 2, $-5\text{ and}-3$ respectively.