Chapter 6, Problem 58SGR

To determine

To solve: each system of equation using an augmented matrix.

Answer to Problem 58SGR

The solution is: $x=22,y=1$ .

Explanation of Solution

Given:

  $\begin{array}{l}2x+3y=8\\ x-5y=-22\end{array}$

Calculation:

The strategy is to perform the indicated operation.

Here, the augmented matrices are:

  $\left[\begin{array}{cc}2& \text{ 3 |}\\ 1& \text{ -5 |}\end{array}\begin{array}{c}\text{ 8}\\ \text{ -22}\end{array}\right]$

Now the matrix in row reduced form will be:

  $\begin{array}{l}\left[\begin{array}{cc}2& \text{ 3 |}\\ 1& \text{ -5 |}\end{array}\begin{array}{c}\text{ 8}\\ \text{ -22}\end{array}\right]\\ \stackrel{R_1-R_2}{\to }\left[\begin{array}{cc}1& \text{ 8 | }\\ 1& \text{ -5 |}\end{array}\begin{array}{c}30\\ \text{ -22}\end{array}\right]\\ \stackrel{R_2-R_1}{\to }\left[\begin{array}{cc}1& \text{ 8 | }\\ 0& \text{ -13 |}\end{array}\begin{array}{c}30\\ \text{ -52}\end{array}\right]\\ \stackrel{R_1-8R_2}{\to }\left[\begin{array}{cc}1& \text{ 0 | }\\ 0& \text{ 1 |}\end{array}\begin{array}{c}22\\ \text{ 1}\end{array}\right]\end{array}$

Conclusion:

Hence, the solution is: $x=22,y=1$ .