Chapter 7.3, Problem 4E

Solution

To identify: The system of equations by using Gaussian elimination.

  $\begin{array}{c}2x-y=0\\ x+3y-z=-3\\ 3y+z=8\end{array}$

The value of $x,y,z$ is

  $\frac{5}{13},\frac{10}{13},\frac{74}{13}$ respectively.

Given information:

The given is.

  $\begin{array}{c}2x-y=0\\ x+3y-z=-3\\ 3y+z=8\end{array}$ .

Explanation:

Consider the given information.

Add the second and third equations, replacing the third equation with the result:

  $\begin{array}{l}\underset{¯}{\begin{array}{c}x+3y-z=-3\\ 3y+z=8\end{array}}\\ x+6y=5\end{array}$

Multiply the third equation by $-2$ and add the result to the first equation, replacing the first equation.

  $\begin{array}{l}\underset{¯}{\begin{array}{c}2x-y=0\\ -2x-12y=-10\end{array}}\\ -13y=-10\end{array}$

So, $y=\frac{10}{13}$

Multiply the first equation by $-6$ and add the result to the third equation, replacing the third equation with the result:

  $\begin{array}{l}\underset{¯}{\begin{array}{l}-6y=-\frac{60}{13}\\ x+6y=5\end{array}}\\ x=\frac{5}{13}\end{array}$

So, $x=\frac{5}{13}$

Substitute the value for $x\text{and}y$ from the first and third equation into the second equation to determine for $z$

  $\begin{array}{c}\frac{5}{13}+3\left(\frac{10}{13}-z\right)=-3\\ z=\frac{74}{13}\end{array}$

So, the value of $x,y,z$ is $\frac{5}{13},\frac{10}{13},\frac{74}{13}$ respectively.