Question

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

$$\begin{gathered} x+y=8 \\ x-y=4 \end{gathered}$$

Short answer

The required solution is$(x,y)=(6,2)$

Step-by-step solution

Step 1. Given Information

The given equations are

$$\begin{gathered} x+y=8 \\ x-y=4 \end{gathered}$$

Step 2. Calculation

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

$\left[\begin{array}{ccc}1& 1|& 8\\ 1& -1|& 4\end{array}\right]$

Now, we will make the entry 0 in row 2 by performing the row operation $R_2=r_2-r_1$and then $R_2=\frac{r_2}{-2}$

localid="1647364426299" $$\begin{gathered} \left[\begin{array}{ccc}1& 1|& 8\\ 1& -1|& 4\end{array}\right]\to \left[\begin{array}{ccc}1& 1|& 8\\ 0& -2|& -4\end{array}\right] \\ \to \left[\begin{array}{ccc}1& 1|& 8\\ 0& 1|& 2\end{array}\right] \end{gathered}$$

Step 3. Calculation

Use the obtained matrix to write the system of equations.

$$\begin{gathered} x+y=8\Rightarrow x=6 \\ y=2 \end{gathered}$$