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Solve using augmented matrix methods: 2x₁ - x₂ = 5 x₁ + 3x₂ = 6 Select the correct choice below and, if...

Solve using augmented matrix methods:
2x₁ - x₂ = 5
x₁ + 3x₂ = 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 answers.)
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.)
C. There is no solution

Answer

Solution:

  1. Convert the system of equations into an augmented matrix:

 [ 2 -1 | 5 ] [ 1  3 | 6 ] 
  1. Use row operations to simplify:

- Multiply the second row by 2 and subtract from the first row:

 [ 2 -1 | 5 ] [ 2  6 | 12 ] 
Subtracting:
 [ 0 -7 | -7 ] [ 1  3 | 6 ] 
  1. Continue simplifying:

- Divide the first row by -7:

 [ 0  1 | 1 ] [ 1  3 | 6 ] 
- Subtract 3 times the first row from the second row:
 [ 0  1 | 1 ] [ 1  0 | 3 ] 
  1. Read the solution from the matrix:

- x₂ = 1 - x₁ = 3

  1. Conclusion:

A. The unique solution is x₁ = 3 and x₂ = 1.

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