Chapter 6.8, Problem 50SR

To determine

each system by using matrices.

Answer to Problem 50SR

The solution is $x=3,y=-4$

Explanation of Solution

Given:

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

Calculation:

Here the equations are:

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

The strategy is to solve the system by using matrices.

Here, the augmented matrices are:

  $\left[\begin{array}{cc}-2& 5\\ 2& -4\end{array}\right]\left[\begin{array}{c}-9\\ 6\end{array}\right]$

Now the matrix in row reduced form will be:

  $\begin{array}{l}\left[\begin{array}{cc}2& 1\\ 5& 2\end{array}\right]\left[\begin{array}{c}2\\ 7\end{array}\right]\\ \stackrel{R_1\leftrightarrow R_2}{\to }\left[\begin{array}{cc}5& 2\\ 2& 1\end{array}\right]\left[\begin{array}{c}7\\ 2\end{array}\right]\\ \stackrel{R_1-2R_2}{\to }\left[\begin{array}{cc}1& 0\\ 2& 1\end{array}\right]\left[\begin{array}{c}3\\ 2\end{array}\right]\\ \stackrel{R_2-2R_1}{\to }\left[\begin{array}{cc}1& 0\text{ |}\\ 0& 1\text{ |}\end{array}\begin{array}{c}3\\ -4\end{array}\right]\end{array}$

Hence, the solution is $x=3,y=-4$

Conclusion:

Hence, the solution is $x=3,y=-4$