Question

Use an augmented matrix to solve each system of equations.

$$\begin{gathered} 3x+5y=7 \\ 2x-3y=11 \end{gathered}$$

Short answer

The solution of the given system of equations is $\left(4,-1\right)$.

Step-by-step solution

Step 1. Write the augmented matrix for the given system of equations.

The given system of equations is:

$$\begin{gathered} 3x+5y=7 \\ 2x-3y=11 \end{gathered}$$

Place the coefficients of the equations and the constant terms into the matrix to form the augmented matrix for the given system of equations.

$\left[\begin{array}{ccc}3& 5& 7\\ 2& -3& 11\end{array}\right]$

Step 2. Solve the given system of equations.

The augmented matrix is:

$\left[\begin{array}{ccc}3& 5& 7\\ 2& -3& 11\end{array}\right]$

Apply $R_1\to \frac{1}{3}{R}_1$,

$\left[\begin{array}{ccc}1& \frac{5}{3}& \frac{7}{3}\\ 2& -3& 11\end{array}\right]$

Apply $R_2\to R_2-2R_1$,

$\left[\begin{array}{ccc}1& \frac{5}{3}& \frac{7}{3}\\ 0& -\frac{19}{3}& \frac{19}{3}\end{array}\right]$

Apply $R_2\to -\frac{3}{19}{R}_2$,

$\left[\begin{array}{ccc}1& \frac{5}{3}& \frac{7}{3}\\ 0& 1& -1\end{array}\right]$

Apply $R_1\to R_1-\frac{5}{3}{R}_2$,

$\left[\begin{array}{ccc}1& 0& 4\\ 0& 1& -1\end{array}\right]$

Therefore, the solution of the given system of equations is $\left(4,-1\right)$.