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Solve using augmented matrix methods. Use a graphing calculator to perform the row operations. 0.16x₁ +...

Solve using augmented matrix methods. Use a graphing calculator to perform the row operations.
0.16x₁ + 2.15x₂ = 6
6.25x₁ + 4.02x₂ = 4
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 write the augmented matrix:

Augmented Matrix:

[ 0.16 2.15 | 6 ]
[ 6.25 4.02 | 4 ]

Perform row operations to reduce the matrix to row-echelon form:

  1. Multiply the first row by 6.25 to make the leading coefficient of the first row equal to the leading coefficient of the second row:

[ 1 13.4375 | 37.5 ]
[ 6.25 4.02 | 4 ]

  1. Subtract 6.25 times the first row from the second row to eliminate the first element of the second row:

[ 1 13.4375 | 37.5 ]
[ 0 -80.6875 | -232.5 ]

Now, solve the system of equations:

x₁ + 13.4375x₂ = 37.5
-80.6875x₂ = -232.5

From the second equation:

x₂ = -232.5 / -80.6875 ≈ 2.8810

Substitute x₂ back into the first equation:

x₁ + 13.4375(2.8810) = 37.5
x₁ ≈ 37.5 - 38.7031
x₁ ≈ -1.2031

Answer:

A. The unique solution is x₁ = -1.2031 and x₂ = 2.8810 (rounded to four decimal places).

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