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Solve using augmented matrix methods. -2x1 + 8x2 = 2 4x1 - 16x2 = -4 Select the correct choice below and,...

Solve using augmented matrix methods.
-2x1 + 8x2 = 2
4x1 - 16x2 = -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 = [ ] (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:

[ -2 8 | 2 ]
[ 4 -16 | -4 ]

We perform row operations to simplify:

  1. Multiply the first row by 2 and add to the second row to eliminate x1 from the second equation:

[ -2 8 | 2 ]
[ 0 0 | 0 ]

This simplifies to:

-2x1 + 8x2 = 2

The second row becomes 0 = 0, which is always true, indicating that the system is consistent and dependent.

Therefore, the system has infinitely many solutions. We express x1 in terms of x2:

-2x1 + 8x2 = 2
x1 = 4x2 - 1

Let x2 = t, then:

x1 = 4t - 1
x2 = t

Answer: B. The system has infinitely many solutions. The solution is x1 = 4t - 1 and x2 = t.

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