Chapter 7.3, Problem 7E

Solution

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

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

The value of $x,y,z,w$ is $\frac{9}{2},\frac{7}{2},4,-\frac{15}{2}$ respectively.

Given information:

The given is.

  $\begin{array}{c}x+y-z=4\\ y+w=-4\\ x-y=1\\ x+z+w=1\end{array}$ .

Explanation:

Consider the given information.

Add the first and third equations to the fourth equation, replacing the fourth equation with result:

  $\begin{array}{l}\underset{¯}{\begin{array}{l}x+y-z=4\\ x+z+w=1\end{array}}\\ 2x+y+w=5\end{array}$

Multiply the first equation by $-1$ and add the result to the fourth equation, replacing the fourth equation.

  $\begin{array}{l}\underset{¯}{\begin{array}{c}-y-w=4\\ 2x+y+w=5\end{array}}\\ 2x=9\end{array}$

So, $x=\frac{9}{2}$

Multiply the third equation by 5 and add the result to the second equation:

  $\begin{array}{l}\underset{¯}{\begin{array}{l}x-y=1\\ -x=-\frac{9}{2}\end{array}}\\ y=-\frac{7}{2}\end{array}$

So, $y=-\frac{7}{2}$

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

  $\begin{array}{l}\underset{¯}{\begin{array}{l}y+w=-4\\ -y=-\frac{7}{2}\end{array}}\\ w=-\frac{15}{2}\end{array}$

So, $w=-\frac{15}{2}$

Substitute the value for $x\text{and}y$ into the first equation to solve for $z$

  $\begin{array}{c}\frac{9}{2}+\frac{7}{2}-z=4\\ 8-z=4\\ z=4\end{array}$

So, the value of $x,y,z,w$ is $\frac{9}{2},\frac{7}{2},4,-\frac{15}{2}$ respectively.