Q. 42 - 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.  

$$\begin{gathered} 3x-y=7 \\ 9x-3y=21 \end{gathered}$$

Step-by-Step Solution

Answer

The required solution is $x=\frac{7}{3}+\frac{y}{3}, y be any real number$

Step 1. Given Information

The given equations are 

$$\begin{gathered} 3x-y=7 \\ 9x-3y=21 \end{gathered}$$

Step 2. Calculation

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

$\left[\begin{array}{ccc}3& -1|& 7\\ 9& -3|& 21\end{array}\right]$

Now, we will make the entry 0 in row 2 column 1 by performing the row operations $R_1=\frac{R_1}{3}$ and then apply the operation $R_2=-9R_1+R_2$ 

$$\begin{gathered} \left[\begin{array}{ccc}3& -1|& 7\\ 9& -3|& 21\end{array}\right]=\left[\begin{array}{ccc}\frac{3}{3}& \frac{-1}{3}|& \frac{7}{3}\\ 9& -3|& 21\end{array}\right] \\ =\left[\begin{array}{ccc}1& \frac{-1}{3}|& \frac{7}{3}\\ 9& -3|& 21\end{array}\right] \\ =\left[\begin{array}{ccc}1& \frac{-1}{3}|& \frac{7}{3}\\ 9-9& -3-9(-\frac{1}{3})|& 21-9\left(\frac{7}{3}\right)\end{array}\right] \\ =\left[\begin{array}{ccc}1& \frac{-1}{3}|& \frac{7}{3}\\ 0& 0|& 0\end{array}\right] \end{gathered}$$

Step 3. Calculation

Use the obtained matrix to write the system of equations.

$x-\frac{y}{3}=\frac{7}{3}$

Solve the equations to find the solution set.

$x=\frac{7}{3}+\frac{y}{3}, y\mathit{ }\mathrm{be} \mathrm{any} \mathrm{real} \mathrm{number}$