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Solve using augmented matrix methods. Use a graphing calculator to perform the row operations 0.8x₁ +...

Solve using augmented matrix methods. Use a graphing calculator to perform the row operations
0.8x₁ + 2.08x₂ = 6
1.25x₁ + 4.14x₂ = 8
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 = [ ] (Round to four decimal places as needed.)
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

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

Augmented Matrix:

[ 0.8 2.08 | 6 ]
[ 1.25 4.14 | 8 ]

Perform row operations to obtain row-echelon form:

  1. Multiply the first row by 1.25 and the second row by 0.8 to eliminate x₁ from the second equation:

[ 1 2.6 | 7.5 ]
[ 1 3.312 | 6.4 ]

  1. Subtract the first row from the second row:

[ 1 2.6 | 7.5 ]
[ 0 0.712 | -1.1 ]

Now, solve the system:

From the second row: 0.712x₂ = -1.1
x₂ = -1.1 / 0.712 ≈ -1.5449

Substitute x₂ back into the first equation:

0.8x₁ + 2.08(-1.5449) = 6
0.8x₁ - 3.215392 = 6
0.8x₁ = 9.215392
x₁ = 9.215392 / 0.8 ≈ 11.5192

Answer: The unique solution is x₁ ≈ 11.5192 and x₂ ≈ -1.5449.

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