Chapter 8.3, Problem 58E

To determine

The system of linear equations by using inverse matrix.

Answer to Problem 58E

  $\left(\begin{array}{c}x=-1+2z\\ y=2-3z\\ z=z\end{array}\right)$ .

Explanation of Solution

Given information:

The given system of equation is:

  $\{\begin{array}{l}2x+3y+5z=4\\ 3x+5y+9z=7\\ 5x+9y+17z=13\end{array}$

Calculation:

The given system of equation is:

  $\{\begin{array}{l}2x+3y+5z=4\\ 3x+5y+9z=7\\ 5x+9y+17z=13\end{array}$

Convert the augmented matrix into the row-echelon form:

Divide the first row by $\frac{1}{2}$ :

  $\left[\begin{array}{ccc}2& 3& 5\\ 3& 5& 9\\ 5& 9& 17\end{array}|\begin{array}{c}4\\ 7\\ 13\end{array}\right]\Rightarrow \left[\begin{array}{ccc}1& 3/2& 5/2\\ 3& 5& 9\\ 5& 9& 17\end{array}|\begin{array}{c}2\\ 7\\ 13\end{array}\right]$

Multiply the first row by three and after that subtract the first row from the second:

  $\left[\begin{array}{ccc}1& 3/2& 5/2\\ 0& 1/2& 3/2\\ 5& 9& 17\end{array}|\begin{array}{c}2\\ 1\\ 13\end{array}\right]$

Multiply the first row by five and after that subtract the first row from the third row:

  $\left[\begin{array}{ccc}1& 3/2& 5/2\\ 0& 1/2& 3/2\\ 0& 3/2& 9/2\end{array}|\begin{array}{c}2\\ 1\\ 3\end{array}\right]$

Divide the second row by $\frac{1}{2}$ .

  $\left[\begin{array}{ccc}1& 3/2& 5/2\\ 0& 1& 3\\ 0& 3/2& 9/2\end{array}|\begin{array}{c}2\\ 2\\ 3\end{array}\right]$

Multiply the second row by $\frac{3}{2}$ after that subtract the second row from the third row:

  $\left[\begin{array}{ccc}1& 3/2& 5/2\\ 0& 1& 3\\ 0& 0& 0\end{array}|\begin{array}{c}2\\ 2\\ 0\end{array}\right]$

Multiply the second row by $\frac{3}{2}$ after that subtract the second row from the first row:

  $\left[\begin{array}{ccc}1& 0& -2\\ 0& 1& 3\\ 0& 0& 0\end{array}|\begin{array}{c}-1\\ 2\\ 0\end{array}\right]$

Now find the variable $x$ from the equation first of the system:

  $x=-1+2z$

Now find the variable $y$ from the equation second of the system:

  $y=2-3z$

Therefore, the solutions are $\left(\begin{array}{c}x=-1+2z\\ y=2-3z\\ z=z\end{array}\right)$