Chapter 8.1, Problem 91E

To determine

Comparing Solutions of Two Systems.

Answer to Problem 91E

System of linear equations has same solution.

  $\boxed{\begin{array}{l}x=-1\\ y=1\\ z=-3\end{array}}$

Explanation of Solution

Given information: determine whether the two systems of linear equations yield the same solution. If so, find the solution using matrices..

a) $\{\begin{array}{ccccc}x& -2y& +z& =& -6\\ & y& -5z& =& 16\\ & & z& =& -3\end{array}$

b) $\{\begin{array}{ccccc}x& +y& -2z& =& 6\\ & y& +3z& =& -8\\ & & z& =& -3\end{array}$

Calculation:

Now the same solution is applied on given system of equations,

Apply augmented matrices operation and then use elementary row operation to reduce the matrix and observe the results of both system of equations.

1. The given system of equations is.

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

The augmented matrix of above system of equations is,

  $\left[\begin{array}{ccccc}1& -2& 1& ⋮& -6\\ 0& 1& -5& ⋮& 16\\ 0& 0& 1& ⋮& -3\end{array}\right]$

Now use Gauss-Jordan elimination method.

  $2R_2+R_1\to R_1$

  $\left[\begin{array}{ccccc}1& 0& -9& ⋮& 26\\ 0& 1& -5& ⋮& 16\\ 0& 0& 1& ⋮& -3\end{array}\right]$

  $\begin{array}{l}9R_3+R_1\to R_1\\ 5R_3+R_2\to R_2\end{array}$

  $\left[\begin{array}{ccccc}1& 0& 0& ⋮& -1\\ 0& 1& 0& ⋮& 1\\ 0& 0& 1& ⋮& -3\end{array}\right]$

b)The given system of equations is.

  $\{\begin{array}{ccccc}x& +y& -2z& =& 6\\ & y& +3z& =& -8\\ & & z& =& -3\end{array}$

The augmented matrix of above system of equations is,

  $\left[\begin{array}{ccccc}1& 1& -2& ⋮& 6\\ 0& 1& 3& ⋮& -8\\ 0& 0& 1& ⋮& -3\end{array}\right]$

Now use Gauss-Jordan elimination method.

  $-R_2+R_1\to R_1$

  $\left[\begin{array}{ccccc}1& 0& -5& ⋮& 14\\ 0& 1& 3& ⋮& -8\\ 0& 0& 1& ⋮& -3\end{array}\right]$

  $\begin{array}{l}5R_3+R_1\to R_1\\ -3R_3+R_2\to R_2\end{array}$

  $\left[\begin{array}{ccccc}1& 0& 0& ⋮& -1\\ 0& 1& 0& ⋮& 1\\ 0& 0& 1& ⋮& -3\end{array}\right]$

The reduced matrices are same, so that the two system of linear equation yield the same solution

Hence there is same solution.

  $\begin{array}{l}x=-1\\ y=1\\ z=-3\end{array}$