Question

Solve each system of equations using matrices (row operations). If the system has no solution, say that it is inconsistent.

$$\begin{gathered} 2x+3y=6 \\ x-y=\frac{1}{2} \end{gathered}$$

Short answer

The required solution is$(\frac{3}{2},1)$

Step-by-step solution

Step 1. Given Information 

The given equations are

$$\begin{gathered} 2x+3y=6 \\ x-y=\frac{1}{2} \end{gathered}$$

Step 2. Calculation 

We will first write the augmented matrix, by writing constants on right side of vertical bar.

$\left[\begin{array}{ccc}2& 3|& 6\\ 1& -1|& \frac{1}{2}\end{array}\right]$

Now, we will make the entry 0 in row 2 column 1 by interchanging rows 1 and 2 and then by performing the row operations $R_2=R_2-R_1$and then apply the operation $R_2=\frac{R_2}{5}$

$$\begin{gathered} \left[\begin{array}{ccc}2& 3|& 6\\ 1& -1|& \frac{1}{2}\end{array}\right]=\left[\begin{array}{ccc}1& -1|& \frac{1}{2}\\ 2& 3|& 6\end{array}\right] \\ =\left[\begin{array}{ccc}1& -1|& \frac{1}{2}\\ 0& 5|& 5\end{array}\right] \\ =\left[\begin{array}{ccc}1& -1|& \frac{1}{2}\\ 0& \frac{5}{5}|& \frac{5}{5}\end{array}\right] \\ =\left[\begin{array}{ccc}1& -1|& \frac{1}{2}\\ 0& 1|& 1\end{array}\right] \end{gathered}$$

Step 3. Calculation 

Use the obtained matrix to write the system of equations.

$$\begin{gathered} x-y=\frac{1}{2} \\ y=1 \end{gathered}$$

Solve the equations to find the solution set.

$$\begin{gathered} y=1 \\ x-1=\frac{1}{2} \\ x=\frac{3}{2} \end{gathered}$$