Question

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

$$\begin{gathered} x+2y=4 \\ 2x+4y=8 \end{gathered}$$

Short answer

The required solution is$\left\{(x,y)|x=4-2y,yisanyrealnumber\right\}$

Step-by-step solution

Step 1. Given Information 

The given equations are

$$\begin{gathered} x+2y=4 \\ 2x+4y=8 \end{gathered}$$

Step 2. Calculation   

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

$\left[\begin{array}{ccc}1& 2|& 4\\ 1& 2|& 4\end{array}\right]$

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

$\left[\begin{array}{ccc}1& 2|& 4\\ 1& 2|& 4\end{array}\right]=\left[\begin{array}{ccc}1& 2|& 4\\ 0& 0|& 0\end{array}\right]$

Step 3. Calculation    

Use the obtained matrix to write the system of equations.

$$\begin{gathered} x+2y=4 \\ 0=0 \end{gathered}$$

Solve the equations to find the solution set.

$$\begin{gathered} x+2y=4 \\ x=4-2y,y\mathrm{is}\mathrm{any}\mathrm{real}\mathrm{number} \end{gathered}$$