Chapter 7, Problem 97RE

To determine

To reduce: the augmented matrix corresponding to the system of equations and solve the system by using matrix capabilities of a graphing utility.

Answer to Problem 97RE

The solution of the system of equations is

  $\left(3,0,-4\right)$ .

Explanation of Solution

Given information:

Given system of equations

  $\{\begin{array}{l}x+2y-z=7\\ -y-z=4\\ 4x-z=16\end{array}$

Calculation:

The augmented matrix corresponding to this system of linear equation is

  $A=\left(\begin{array}{cccc}1& 2& -1& 7\\ 0& -1& -1& 4\\ 4& 0& -1& 16\end{array}\right)$

Now, using the matrix capabilities of graphing utility, we can reduce augmented matrix of $A$ as follows:

Press $2^{nd}$ and matrix keys, then edit key.

Now enter the elements of matrix.

1, 2, 3 columns are shown below:

  

4^th column is shown below:

  

The echelon form is

  

Converting decimals to rationals, we get

  $\left(\begin{array}{cccc}1& 2& -1& 7\\ 0& -1& -1& 4\\ 0& 0& 11& -44\end{array}\right)$

Now we write down corresponding triangular system

  $\{\begin{array}{l}x+2y-z=7\\ -y-z=4\\ 11z=-44\end{array}$

By back substitution we have

  $\begin{array}{c}z=\frac{-44}{11}\\ =-4\\ -y-z=4\\ -y-\left(-4\right)=4\\ -y+4=4\\ y=0\end{array}$

and

  $\begin{array}{c}x+2y-z=7\\ x+2\cdot 0-\left(-4\right)=7\\ x+0+4=7\\ x=7-4\\ x=3\end{array}$

Therefore the solution of the system of equations is

  $\left(3,0,-4\right)$ .