Chapter 8.1, Problem 59E

To determine

To solve:The system of linear equations using Gaussian elimination method with back-substitution.

Answer to Problem 59E

Thesolution of the given system is $x=-3\text{and}y=5$ .

Explanation of Solution

Given information:

The system of equations is

  $\{\begin{array}{c}x+2y=7\\ -x+y=8\end{array}$

Calculations:

Writing the given system of linear equations in the augmented matrix form, we have

  $\left[\begin{array}{cccc}1& 2& :& 7\\ -1& 1& :& 8\end{array}\right]$

Performing the row operations to bring it in row-echelon form, we get

  $\begin{array}{l}\left[\begin{array}{cccc}1& 2& :& 7\\ 0& 3& :& 15\end{array}\right]\begin{array}{c}\\ \leftarrow \end{array}\begin{array}{c}\\ R_2+R_1\end{array}\\ \left[\begin{array}{cccc}1& 2& :& 7\\ 0& 1& :& 5\end{array}\right]\begin{array}{c}\\ \leftarrow \end{array}\begin{array}{c}\\ R_2/3\end{array}\end{array}$

Now, the matrix is in row-echelon form, and writing it in the system form, we have

  $\{\begin{array}{c}x+2y=7\\ y=5\end{array}$

Substituting $y=5$ in the first equation, we get

  $\begin{array}{l}x+2(5)=7\\ \to x+10=7\\ \Rightarrow x=-3\end{array}$

Conclusion:

Therefore, the solution of the given system is $x=-3\text{and}y=5$ .