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Solve using augmented matrix methods. 2x₁ + 8x₂ = 4 -3x₁ - 12x₂ = -6 Select the correct choice below and,...

Solve using augmented matrix methods.
2x₁ + 8x₂ = 4
-3x₁ - 12x₂ = -6
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The unique solution is x₁ = ___ and x₂ = ___ (Simplify your answer.)
B. The system has infinitely many solutions. The solution is x₁ = ___ and x₂ = t. (Simplify your answer. Type an expression using t as the variable. Do not factor.)
C. There is no solution.

Answer

First, write the system of equations as an augmented matrix:

[ [2, 8, 4], [-3, -12, -6] ]

Perform row operations to simplify the matrix:

  1. Multiply the first row by 3: [ [6, 24, 12], [-3, -12, -6] ]
  2. Add the second row to the first row: [ [3, 12, 6], [-3, -12, -6] ]
  3. Multiply the first row by 2: [ [6, 24, 12], [-3, -12, -6] ]
  4. Add the second row to the first row: [ [0, 0, 0], [-3, -12, -6] ]

The resulting matrix indicates that the system has infinitely many solutions.

Answer: B. The system has infinitely many solutions. The solution is x₁ = -4t and x₂ = t.

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