Questions: x-2y+z=-4 -4x+y-2z=1 2x+2y-z=10

x-2y+z=-4 -4x+y-2z=1 2x+2y-z=10
Transcript text: $\begin{array}{c}x-2 y+z=-4 \\ -4 x+y-2 z=1 \\ 2 x+2 y-z=10\end{array}$
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Solution

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Solution Steps

Step 1: Matrix Representation

The system of equations is represented in matrix form as \(AX = B\), where: $$ \begin{align_} A &= \begin{bmatrix} 1 & -2 & 1 \\ -4 & 1 & -2 \\ 2 & 2 & -1 \end{bmatrix}, X &= \begin{bmatrix} x \\ y \\ z \end{bmatrix}, B &= \begin{bmatrix} -4 \\ 1 \\ 10 \end{bmatrix} \end{align_} $$

Step 2: Determinant Check

The determinant of matrix \(A\) is 9.

Step 3: Solution Method

Since the determinant is non-zero, the system has a unique solution. The solution is found using the inverse of matrix \(A\) to find \(X = A^{-1}B\). The solution is \(x = 2, y = 1, z = -4\).

Final Answer:

The unique solution to the system of equations is \(x = 2, y = 1, z = -4\).

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