Question

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

$$\begin{gathered} \frac{1}{2}x+y=-2 \\ x-2y=8 \end{gathered}$$

Short answer

The required solution is$x=2,y=-3$

Step-by-step solution

Step 1. Given Information 

The given equations are

$$\begin{gathered} \frac{1}{2}x+y=-2 \\ x-2y=8 \end{gathered}$$

Step 2. Calculation 

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

$\left[\begin{array}{ccc}\frac{1}{2}& 1|& -2\\ 1& -2|& 8\end{array}\right]$

Now, we will make the entry 0 in row 2 column 1 by performing the row operations $R_1=2R_1$and then apply the operation $R_2=-R_1+R_2$

$$\begin{gathered} \left[\begin{array}{ccc}\frac{1}{2}& 1|& -2\\ 1& -2|& 8\end{array}\right]=\left[\begin{array}{ccc}2\left(\frac{1}{2}\right)& 2\left(1\right)|& 2(-2)\\ 1& -2|& 8\end{array}\right] \\ =\left[\begin{array}{ccc}1& 2|& -4\\ 1& -2|& 8\end{array}\right] \\ =\left[\begin{array}{ccc}1& 2|& -4\\ -1+1& -2-2|& -(-4)+8\end{array}\right] \\ =\left[\begin{array}{ccc}1& 2|& -4\\ 0& -4|& 12\end{array}\right] \end{gathered}$$

Step 3. Calculation 

Use the obtained matrix to write the system of equations.

$$\begin{gathered} x+2y=-4 \\ -4y=12 \end{gathered}$$

Solve the equations to find the solution set.

$$\begin{gathered} -4y=12\Rightarrow y=-3 \\ x+2(-3)=-4 \\ x-6=-4 \\ x=-4+6 \\ x=2 \end{gathered}$$