Q. 39 - 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.  

$$\begin{gathered} 2x-4y=-2 \\ 3x+2y=3 \end{gathered}$$

Step-by-Step Solution

Answer

The required solution is $(x,y)=(\frac{1}{2},\frac{3}{4})$

Step 1. Given Information

The given equations are  

$$\begin{gathered} 2x-4y=-2 \\ 3x+2y=3 \end{gathered}$$

Step 2. Calculation

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

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

Now, we will make the entry 0 in row 2 column 1 by performing the row operation $R_2=r_2-3r_1$ and then apply $R_2=\frac{r_2}{8}$ 

$$\begin{gathered} \left[\begin{array}{ccc}1& -2|& -1\\ 3& 2|& 3\end{array}\right]=\left[\begin{array}{ccc}1& -2|& -1\\ 0& 8|& 6\end{array}\right] \\ =\left[\begin{array}{ccc}1& -2|& -1\\ 0& 1|& \frac{6}{8}\end{array}\right] \end{gathered}$$

Step 3. Calculation

Use the obtained matrix to write the system of equations.

$$\begin{gathered} x-2y=-1 \\ y=\frac{6}{8}=\frac{3}{4} \end{gathered}$$

Solve the equations to find the solution set.

$$\begin{gathered} y=\frac{3}{4} \\ x-2\left(\frac{3}{4}\right)=-1 \\ x=\frac{1}{2} \end{gathered}$$