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Solve using augmented matrix methods: -8x₁ + 4x₂ = 12 6x₁ - 3x₂ = -9 Select the correct choice below and,...

Solve using augmented matrix methods:
-8x₁ + 4x₂ = 12
6x₁ - 3x₂ = -9
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 = [ ]. (Simplify your answer. Type an expression using t as the variable.)
C. There is no solution.

Answer

To solve the system using augmented matrix methods, we first write the augmented matrix:

Augmented Matrix:

[ -8 4 | 12 ]
[ 6 -3 | -9 ]

Perform row operations to simplify:

  1. Multiply the second row by 4 and the first row by 3 to make the coefficients of x₂ equal:

[ -24 12 | 36 ]
[ 24 -12 | -36 ]

  1. Add the two rows:

[ 0 0 | 0 ]

The resulting row indicates that the system is dependent, leading to infinitely many solutions.

Answer: B. The system has infinitely many solutions. The solution is x₁ = t and x₂ = 2t - 3, where t is a parameter.

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