Questions: Solve the system. -3x-y=6 y-2z=1 x-4y+z=9 Select the correct choice below and fill in any answer boxes within your choice. A. There is one solution. The solution set is ( , , ) (Simplify your answers.) B. There are infinitely many solutions. C. There is no solution.

Solve the system. -3x-y=6 y-2z=1 x-4y+z=9 Select the correct choice below and fill in any answer boxes within your choice. A. There is one solution. The solution set is ( , , ) (Simplify your answers.) B. There are infinitely many solutions. C. There is no solution.
Transcript text: Solve the system. \[ \begin{array}{r} -3 x-y=6 \\ y-2 z=1 \\ x-4 y+z=9 \] Select the correct choice below and fill in any answer boxes within your choice. A. There is one solution. The solution set is $\{($ $\square$ $\square$ , $\square$ ) \} (Simplify your answers.) B. There are infinitely many solutions. C. There is no solution.
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Solution

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Solution Steps

Step 1: Formulate the System of Equations

We start with the given system of equations: \[ \begin{array}{r} -3x - y = 6 \\ y - 2z = 1 \\ x - 4y + z = 9 \end{array} \]

Step 2: Represent the System in Matrix Form

We can represent this system in the augmented matrix form \( [A | b] \): \[ \left[ A | b \right] = \left[ \begin{array}{ccc|c} -3 & -1 & 0 & 6 \\ 0 & 1 & -2 & 1 \\ 1 & -4 & 1 & 9 \\ \end{array} \right] \]

Step 3: Perform Gaussian Elimination

Through Gaussian elimination, we transform the matrix step by step until we reach the reduced row echelon form: \[ \left[ A | b \right] = \left[ \begin{array}{ccc|c} 1 & 0 & 0 & -1 \\ 0 & 1 & 0 & -3 \\ 0 & 0 & 1 & -2 \\ \end{array} \right] \]

Step 4: Extract the Solutions

From the final matrix, we can read off the solutions for the variables: \[ x = -1, \quad y = -3, \quad z = -2 \]

Final Answer

The solution set is \(\{(x, y, z) = (-1, -3, -2)\}\). Therefore, the answer is A. \[ \boxed{(x, y, z) = (-1, -3, -2)} \]

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