Questions: Consider the following system. -3x + y + 2z = 4 5x - y - z = -1 -4x + 2y + 5z = 9 Choose the best description of its solution. If applicable, give its solution. The system has no solution. The system has a unique solution. (x, y, z) = The system has infinitely many solutions. (x, y, z) = (x, , ) (ฤ, y, ) (, , z)

Solution Steps

To solve the given system of linear equations, we can use matrix methods such as Gaussian elimination or matrix inversion if the matrix is invertible. The goal is to find the values of \(x\), \(y\), and \(z\) that satisfy all three equations simultaneously. We will represent the system in matrix form \(AX = B\) and solve for \(X\).

We are given the following system of equations: \[ \begin{align_} -3x + y + 2z &= 4 \quad (1) \\ 5x - y - z &= -1 \quad (2) \\ -4x + 2y + 5z &= 9 \quad (3) \end{align_} \]

Using matrix methods, we represent the system in the form \(AX = B\), where: \[ A = \begin{bmatrix} -3 & 1 & 2 \\ 5 & -1 & -1 \\ -4 & 2 & 5 \end{bmatrix}, \quad B = \begin{bmatrix} 4 \\ -1 \\ 9 \end{bmatrix} \] The solution to the system yields: \[ \begin{align_} x &\approx -2.0471 \times 10^{15} \\ y &\approx -1.4330 \times 10^{16} \\ z &\approx 4.0942 \times 10^{15} \end{align_} \]

The values obtained for \(x\), \(y\), and \(z\) are extremely large in magnitude, indicating that the system does not have a unique solution. Instead, it suggests that the system may be inconsistent or dependent, leading to either no solution or infinitely many solutions.

Final Answer