Chapter 11, Problem 13CT

To determine

The solution of the system of equations $\{\begin{array}{l}6x+3y=1\\ 2x-y=-2\end{array}$ by using matrices. If system has no solution then state it is inconsistent.

Answer to Problem 13CT

Solution:

The solution of the system of equations $\{\begin{array}{l}6x+3y=1\\ 2x-y=-2\end{array}$ is $x=\frac{1}{2},y=3$ or $\left(\frac{1}{2},3\right)$.

Explanation of Solution

Given information:

The system of equations $\{\begin{array}{l}6x+3y=1\\ 2x-y=-2\end{array}$.

Explanation:

The augmented matrix of the system is,

$\left[\begin{array}{cc}6& 3\\ 2& -1\end{array}|\begin{array}{c}12\\ -2\end{array}\right]$

Perform the row operation $R_1\leftrightarrow R_2$,

$\Rightarrow \left[\begin{array}{cc}2& -1\\ 6& 3\end{array}|\begin{array}{c}-2\\ 12\end{array}\right]$

Perform the row operation $R_1\to \frac{1}{2}{r}_1$, $\Rightarrow \left[\begin{array}{cc}1& -\frac{1}{2}\\ 6& 3\end{array}|\begin{array}{c}-1\\ 12\end{array}\right]$

Perform the row operation $R_2\to -6r_1+r_2$,

$\Rightarrow \left[\begin{array}{cc}1& -\frac{1}{2}\\ 0& 6\end{array}|\begin{array}{c}-1\\ 18\end{array}\right]$

Perform the row operation $R_2\to \frac{1}{6}{R}_2$

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

Perform the row operation $R_2\to \frac{1}{2}{r}_2+r_1$,

$\Rightarrow \left[\begin{array}{cc}1& 0\\ 0& 1\end{array}|\begin{array}{c}\frac{1}{2}\\ 3\end{array}\right]$

Hence, the solution of system of equation $\{\begin{array}{l}6x+3y=1\\ 2x-y=-2\end{array}$ is $x=\frac{1}{2},y=3$ or using an ordered pair $\left(\frac{1}{2},3\right)$.