The augmented matrix that corresponds to a system of linear equations in \( x_{1} \) and \( x_{2} \) \( \left[\begin{array}{cc|c}1 & -2 & 9 \\ -3 & 7 & -29\end{array}\right] \) \( x_{1}= \) Ex: 42 \( x_{2}= \)

To solve the augmented matrix, we can use row operations. Starting with the matrix: \[ \begin{bmatrix} 1 & -2 & | & 9 \\ -3 & 7 & | & -29 \end{bmatrix} \] First, let's eliminate \(x_1\) from the second row. We can achieve this by multiplying the first row by 3 and adding it to the second row: \[ 3 \cdot R_1 + R_2 \rightarrow R_2 \] This gives us: \[ \begin{bmatrix} 1 & -2 & | & 9 \\ 0 & 1 & | & -2 \end{bmatrix} \] Now, we can express \(x_2\) directly from the second row: \[ x_2 = -2 \] Next, substituting \(x_2 = -2\) back into the first equation to find \(x_1\): \[ x_1 - 2(-2) = 9 \\ x_1 + 4 = 9 \\ x_1 = 5 \] The final solution is: \(x_{1} = 5\) \\ \(x_{2} = -2\)