Q. 41 - Precalculus Enhanced with Graphing Utilities

Question

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

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

Step-by-Step Solution

Answer

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

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.

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