Questions: Use the Gauss-Jordan method to solve the system of equations. 15) x-y+z=-2 x+y+z=-6 x+y-z=4 A) No solution B) (1,-2,-5) C) (1,-5,-2) D) (-5,1,-2)

Solution Steps

To solve the system of equations using the Gauss-Jordan method, we will first represent the system as an augmented matrix. Then, we will perform row operations to transform the matrix into reduced row-echelon form (RREF). Finally, we will interpret the resulting matrix to find the solution to the system of equations.

The given system of equations is: \[ \begin{align_} x - y + z &= -2 \\ x + y + z &= -6 \\ x + y - z &= 4 \end{align_} \] We can represent this system as an augmented matrix: \[ A = \begin{bmatrix} 1 & -1 & 1 & | & -2 \\ 1 & 1 & 1 & | & -6 \\ 1 & 1 & -1 & | & 4 \end{bmatrix} \]

By applying row operations to transform the augmented matrix into reduced row-echelon form (RREF), we obtain: \[ \begin{bmatrix} 1 & 0 & 0 & | & 1 \\ 0 & 1 & 0 & | & -2 \\ 0 & 0 & 1 & | & -5 \end{bmatrix} \]

From the RREF, we can read the solutions directly: \[ \begin{align_} x &= 1 \\ y &= -2 \\ z &= -5 \end{align_} \] Thus, the solution to the system of equations is \( (1, -2, -5) \).

Final Answer