Chapter 7, Problem 98RE

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 98RE

The solution of the system of equations is

  $\left\{-2,1,1\right\}$ .

Explanation of Solution

Given information:

Given system of equations

  $\{\begin{array}{l}3x+6z=0\\ -2x+y=5\\ y+2z=16\end{array}$

Calculation:

The augmented matrix corresponding to this system of linear equation is

  $A=\left(\begin{array}{cccc}3& 0& -6& 0\\ -2& 1& 0& 5\\ 0& 1& 2& 3\end{array}\right)$

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

Press $2^{nd}$ and matrix keys and then matrix key. Now press edit key. We shall get the matrix.

Enter elements of the matrices. Clear it.Again press 2^nd, matrix keys. Select math key and A:ref(

Finally select names and then press enter key

  

Multiplying 1^st row by 3, 2^nd row by 3 and 3^rd row by $-6$ of resulting matrix, we get

  $\left(\begin{array}{cccc}3& 0& 6& 0\\ 0& 3& 12& 15\\ 0& 0& -6& -6\end{array}\right)$

Now we write down corresponding triangular system

  $\{\begin{array}{l}3x+6z=7\\ 3y+12z=4\\ -6z=-6\end{array}$

By back substitution we have

  $\begin{array}{c}z=\frac{-6}{-6}\\ =1\\ 3y+12z=15\\ 3y+12\cdot 1=15\\ 3y=15-12\\ y=1\end{array}$

and

  $\begin{array}{c}3x+6z=0\\ 3x+6\cdot 1=0\\ x=-2\end{array}$

Therefore the solution of the system of equations is

  $\left\{-2,1,1\right\}$ .