Questions: Using Gauss-Jordan elimination to solve the system x-y+z=-5, 3x+y-4z=15, 4x-5y+8z=-32 , the matrix obtained is [ 1 0 0 1; 0 1 0 4; 0 0 1 -2 ]. Complete the following sentence. The system's solution set is ( 1, 4, -2 ) .

Solution Steps

To solve the system of equations using Gauss-Jordan elimination, we first convert the system into an augmented matrix. Then, we perform row operations to transform the matrix into reduced row-echelon form (RREF). The RREF matrix directly gives us the solutions for the variables. In this case, the RREF matrix is already provided, so we can directly read off the solutions for \(x\), \(y\), and \(z\).

The given system of equations is: \[ \begin{align*}

1. & \quad x - y + z = -5 \\
2. & \quad 3x + y - 4z = 15 \\
3. & \quad 4x - 5y + 8z = -32 \end{align*} \]

The augmented matrix corresponding to the system is: \[ \left[\begin{array}{rrr|r} 1 & -1 & 1 & -5 \\ 3 & 1 & -4 & 15 \\ 4 & -5 & 8 & -32 \end{array}\right] \]

After applying Gauss-Jordan elimination, we arrive at the reduced row-echelon form (RREF): \[ \left[\begin{array}{rrr|r} 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 4 \\ 0 & 0 & 1 & -2 \end{array}\right] \]

From the RREF matrix, we can directly read the solutions: \[ \begin{align_} x & = 1 \\ y & = 4 \\ z & = -2 \end{align_} \]

Final Answer