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Solve using augmented matrix methods: -6x₁ + 2x₂ = 2 9x₁ - 3x₂ = -3 Select the correct choice below and, if...

Solve using augmented matrix methods:
-6x₁ + 2x₂ = 2
9x₁ - 3x₂ = -3
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

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

Augmented Matrix:

[ -6 2 | 2 ]
[ 9 -3 | -3 ]

We perform row operations to simplify:

  1. Multiply the first row by 3: [ -18 6 | 6 ]
  2. Multiply the second row by 2: [ 18 -6 | -6 ]
  3. Add the two rows: [ 0 0 | 0 ]

The resulting system is:

0 = 0

This indicates that the system has infinitely many solutions. We express x₁ in terms of x₂ (let x₂ = t):

Solution:

B. The system has infinitely many solutions. The solution is x₁ = -1/3t + 1/3 and x₂ = t.

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