Q. 45 - Precalculus Enhanced with Graphing Utilities

Question

In Problems 37–72, solve each system of equations using matrices (row operations). If the system has no solution, say that it is inconsistent 

$\left\{\begin{array}{l}3x-5y=3\\ 15x+5y=21\end{array}\right.$

Step-by-Step Solution

Answer

The solution of the system is $x=\frac{4}{3},y=\frac{1}{5}$or, using ordered pair, $\left(\frac{4}{3},\frac{1}{5}\right)$.

Step 1. Given information.

The given system of equation are 

$\left\{\begin{array}{l}3x-5y=3\\ 15x+5y=21\end{array}\right.$

Step 2. Calculation.

The augmented matrix of the system is:

$\left[\begin{array}{cc}\begin{array}{cc}3& -5\\ 15& 5\end{array}& \begin{array}{c}3\\ 21\end{array}\end{array}\right]$

Perform the row operations $R_2=r_2-5r_1$:

$\left[\begin{array}{cc}\begin{array}{cc}3& -5\\ 0& 30\end{array}& \begin{array}{c}3\\ 6\end{array}\end{array}\right]$

Perform the row operations $R_2=\frac{1}{30}{r}_2$:

$\left[\begin{array}{cc}\begin{array}{cc}3& -5\\ 0& 1\end{array}& \begin{array}{c}3\\ \frac{1}{5}\end{array}\end{array}\right]$

Use the obtained matrix to write the system of equations. 

$$\begin{gathered} 3x-5y=3 \\ y=\frac{1}{5} \end{gathered}$$

Step 3. Solve the equation.

Solve the equations to find the solution set. 

Substitute $y=\frac{1}{5}$into the equation.

$$\begin{gathered} 3x-5y=3 \\ 3x-5\left(\frac{1}{5}\right)=3 \\ 3x-1=3 \\ 3x=4 \\ x=\frac{4}{3} \end{gathered}$$

The solution of the system is $x=\frac{4}{3},y=\frac{1}{5}$or, using ordered pair, $\left(\frac{4}{3},\frac{1}{5}\right)$.

Step 4. Conclusion.

The solution of the system is $x=\frac{4}{3},y=\frac{1}{5}$or, using ordered pair, $\left(\frac{4}{3},\frac{1}{5}\right)$.