Chapter 2.5, Problem 34E

To determine

To find: solution of the given system of equations using matrix equation.

Answer to Problem 34E

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

Explanation of Solution

Given information:

The given equations are as follows:

  $\begin{array}{l}4x-y=1\\ x+2y=7\end{array}$

Calculation:

Set up the matrix for the given equations as follows:

  $\begin{array}{l}\begin{array}{ccc}A& B& \text{ C}\end{array}\\ \left[\begin{array}{cc}\begin{array}{l}4\\ 1\end{array}& \begin{array}{l}-1\\ 2\end{array}\end{array}\right]\left[\begin{array}{l}x\\ y\end{array}\right]=\left[\begin{array}{l}1\\ 7\end{array}\right]\end{array}$

For $A=\left[\begin{array}{cc}\begin{array}{l}a\\ c\end{array}& \begin{array}{l}b\\ d\end{array}\end{array}\right]$ , the inverse formula is $A^{-1}=\frac{1}{ad-bc}\left[\begin{array}{cc}\begin{array}{l}d\\ -c\end{array}& \begin{array}{l}-b\\ a\end{array}\end{array}\right]$ .

Put the values in the above formula and evaluate the inverse for the given matrix as follows:

  $\begin{array}{c}{\left[\begin{array}{cc}\begin{array}{l}4\\ 1\end{array}& \begin{array}{l}-1\\ 2\end{array}\end{array}\right]}^{-1}=\frac{1}{8+1}\left[\begin{array}{cc}\begin{array}{l}2\\ -1\end{array}& \begin{array}{l}1\\ 4\end{array}\end{array}\right]\\ =\frac{1}{9}\left[\begin{array}{cc}\begin{array}{l}2\\ -1\end{array}& \begin{array}{l}1\\ 4\end{array}\end{array}\right]\end{array}$

Multiply both the sides of the matrix by the inverse as follows:

  $\frac{1}{9}\left[\begin{array}{cc}\begin{array}{l}2\\ -1\end{array}& \begin{array}{l}1\\ 4\end{array}\end{array}\right]\left[\begin{array}{cc}\begin{array}{l}4\\ 1\end{array}& \begin{array}{l}-1\\ 2\end{array}\end{array}\right]\left[\begin{array}{l}x\\ y\end{array}\right]=\frac{1}{9}\left[\begin{array}{cc}\begin{array}{l}2\\ -1\end{array}& \begin{array}{l}1\\ 4\end{array}\end{array}\right]\left[\begin{array}{l}1\\ 7\end{array}\right]$

Now solve each side to get,

  $\begin{array}{l}\left[\begin{array}{l}x\\ y\end{array}\right]=\frac{1}{9}\left[\begin{array}{l}2+7\\ -1+28\end{array}\right]\\ =\frac{1}{9}\left[\begin{array}{c}9\\ 27\end{array}\right]\\ =\left[\begin{array}{l}1\\ 3\end{array}\right]\end{array}$

Thus, the value of x and y is 1 and 3 respectively.