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Math

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PROBLEM

[100−501−18000−6]\left[\begin{array}{ccc:c} 1 & 0 & 0 & -5 \\ 0 & 1 & -1 & 8 \\ 0 & 0 & 0 & -6 \end{array}\right] The system has no solution.
The system has a unique solution.
(x,y,z)=(□,□,□)(x, y, z)=(\square, \square, \square) The system has infinitely many solutions.
(x,y,z)=(x,□,□)(x, y, z)=(x, \square, \square)
(x,y,z)=(□,y,□)(x, y, z)=(\square, y, \square)
(x,y,z)=(□,□,z)(x, y, z)=(\square, \square, z)

STEP 1

1. The matrix represents an augmented matrix for a system of linear equations.
2. The system of equations is in row-echelon form.
3. We need to determine the nature of the solutions based on the matrix.

STEP 2

1. Analyze the matrix to determine if the system has no solution.
2. Analyze the matrix to determine if the system has a unique solution.
3. Analyze the matrix to determine if the system has infinitely many solutions.

STEP 3

Analyze the matrix to determine if the system has no solution:
The last row of the matrix is [0 0 0 ∣ −6] [0 \, 0 \, 0 \, | \, -6] , which represents the equation 0x+0y+0z=−6 0x + 0y + 0z = -6 . This is a contradiction because 0≠−6 0 \neq -6 .

STEP 4

Since there is a contradiction in the system, it cannot have a unique solution. Therefore, we do not need to analyze for a unique solution.

SOLUTION

Since the system has a contradiction, it cannot have infinitely many solutions either.
The system has no solution.

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