Questions: Solve using augmented matrix methods. 4x1-2x2 =-5 -8x1+4x2 =-11 Select the correct choice below and fill in any answer boxes in your choice. A. The unique solution is x1= and x2= . (Simplify your answer.) B. The system has infinitely many solutions. The solution is x1= and x2=t. (Simplify your answer. Type an expression using t as the variable.) C. There is no solution.

Solution Steps

To solve the given system of linear equations using augmented matrix methods, we first represent the system as an augmented matrix. Then, we perform row operations to bring the matrix to its row-echelon form or reduced row-echelon form. Finally, we interpret the resulting matrix to determine the solution of the system, checking for consistency and the number of solutions.

The given system of equations is: \[ \begin{aligned} 4 x_{1} - 2 x_{2} & = -5 \\ -8 x_{1} + 4 x_{2} & = -11 \end{aligned} \] We can represent this system as an augmented matrix: \[ A = \begin{bmatrix} 4 & -2 & -5 \\ -8 & 4 & -11 \end{bmatrix} \]

We perform row operations to simplify the matrix. By multiplying the first row by \(2\) and adding it to the second row, we obtain: \[ A[1] = A[1] + 2 \cdot A[0] \implies A = \begin{bmatrix} 4 & -2 & -5 \\ 0 & 0 & -21 \end{bmatrix} \]

The resulting matrix indicates that the second row is \(0x_1 + 0x_2 = -21\). This is a contradiction since \(0\) cannot equal \(-21\). Therefore, the system of equations is inconsistent.

Final Answer