Q. 40 - 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.   

$$\begin{gathered} 3x+3y=3 \\ 4x+2y=\frac{8}{3} \end{gathered}$$

Step-by-Step Solution

Answer

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

Step 1. Given Information

The given equations are 

$$\begin{gathered} 3x+3y=3 \\ 4x+2y=\frac{8}{3} \end{gathered}$$

Step 2. Calculation

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

$\left[\begin{array}{ccc}3& 3|& 3\\ 4& 2|& \frac{8}{3}\end{array}\right]$

Now, we will make the entry 0 in row 2 column 1 by performing the row operations $R_1=\frac{1}{3}{R}_1$ and then apply $R_2=-4R_1+R_2$ 

$$\begin{gathered} \left[\begin{array}{ccc}3& 3|& 3\\ 4& 2|& \frac{8}{3}\end{array}\right]=\left[\begin{array}{ccc}\frac{3}{3}& \frac{3}{3}|& \frac{3}{3}\\ 4& 2|& \frac{8}{3}\end{array}\right] \\ =\left[\begin{array}{ccc}1& 1|& 1\\ 4& 2|& \frac{8}{3}\end{array}\right] \\ =\left[\begin{array}{ccc}1& 1|& 1\\ -4\left(1\right)+4& -4\left(1\right)+2|& -4\left(1\right)+\frac{8}{3}\end{array}\right] \\ =\left[\begin{array}{ccc}1& 1|& 1\\ 0& -2|& \frac{-4}{3}\end{array}\right] \end{gathered}$$

Step 3. Calculation

Use the obtained matrix to write the system of equations.

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

Solve the equations to find the solution set.

$$\begin{gathered} y=\frac{-4}{3}\times \frac{-1}{2}\Rightarrow y=\frac{2}{3} \\ x+\frac{2}{3}=1 \\ x=\frac{1}{3} \end{gathered}$$