Q. 38 - 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. 

$$\begin{gathered} x+2y=5 \\ x+y=3 \end{gathered}$$

Step-by-Step Solution

Answer

The required solution is $x=1,y=2$

Step 1. Given Information

The given equations are  

$$\begin{gathered} x+2y=5 \\ x+y=3 \end{gathered}$$

Step 2. Calculation

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

$\left[\begin{array}{ccc}1& 2|& 5\\ 1& 1|& 3\end{array}\right]$

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

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

Step 3. Calculation

Use the obtained matrix to write the system of equations.

$$\begin{gathered} x+2y=5 \\ -y=-2 \end{gathered}$$

Solve the equations to find the solution set.

$$\begin{gathered} y=2 \\ x+2\left(2\right)=5 \\ x+4=5 \\ x=1 \end{gathered}$$