Write the system of equations corresponding to each augmented matrix. Then perform the indicated row operation(s) on the given augmented matrix.

$\left[\begin{array}{cccc}1& -3& 4|& 3\\ 3& -5& 6|& 6\\ -5& 3& 4|& 6\end{array}\right],R_2=-3r_1+r_2,R_3=5r_1+r_3$

The given matrix and operation is

$\left[\begin{array}{cccc}1& -3& 4|& 3\\ 3& -5& 6|& 6\\ -5& 3& 4|& 6\end{array}\right],R_2=-3r_1+r_2,R_3=5r_1+r_3$

The given matrix has three rows. So, it represents a system of three linear equations. The 3 columns to the left of vertical bar indicate that the system has three variables.

Let the variables be x, y and z. Then system of linear equation corresponding to the given matrix is as follows,

$$\begin{gathered} x-3y+4z=3 \\ 3x-5y+6z=6 \\ -5x+3y+4z=6 \end{gathered}$$

Apply the first-row operation on the given augmented matrix and simplify,

localid="1647340869666" $$\begin{gathered} \left[\begin{array}{cccc}1& -3& 4|& 3\\ 3& -5& 6|& 6\\ -5& 3& 4|& 6\end{array}\right]\to \left[\begin{array}{cccc}1& -3& 4|& 3\\ -3\left(1\right)+3& -3(-3)+(-5)& -3\left(4\right)+6|& -3\left(3\right)+6\\ -5& 3& 4|& 6\end{array}\right] \\ \to \left[\begin{array}{cccc}1& -3& 4|& 3\\ 0& 4& -6|& -3\\ -5& 3& 4|& 6\end{array}\right] \end{gathered}$$

Now, apply the second-row operation on the obtained matrix and simplify,

localid="1647504039482" $$\begin{gathered} \left[\begin{array}{cccc}1& -3& 4|& 3\\ 0& 4& -6|& -3\\ -5& 3& 4|& 6\end{array}\right]\to \left[\begin{array}{cccc}1& -3& 4|& 3\\ 0& 4& -6|& -3\\ 5\left(1\right)-5& 5(-3)+3& 5\left(4\right)+4|& 5\left(3\right)+6\end{array}\right] \\ \to \left[\begin{array}{cccc}1& -3& 4|& 3\\ 0& 4& -6|& -3\\ 0& -12& 24|& 21\end{array}\right] \end{gathered}$$