Given that (3, 2, - 6) and (-2, 5, 1) are solutions of two equations in a system of three linear equations, which of the following is true about the system?

a. The system can only be consistent and independent.

b. The system can be either inconsistent and independent or consistent and dependent.

c. The system can be either inconsistent and dependent or consistent and independent.

d. The system can only be inconsistent and dependent.

none of the above

Step-by-step explanation:

The system can be any of ...

consistent and independent inconsistent dependent

"Inconsistent" means there is no solution. "Dependent" means there are infinite solutions. It is not possible to be "inconsistent and dependent." (This leaves out choices b, c, and d.

Below, we show that "consistent and independent" is not the only possibility, leaving out choice a.

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Consistent and independent

x + y + 2 z = - 7 (3, 2, - 6) is a solution x + 2 y + z = 9 (-2, 5, 1) is a solution 2 x + y + z = 2

The one solution to this set of equations is (x, y, z) = (1, 8, - 8).

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Inconsistent

1024x + 289y + 603z = 32 (3, 2, - 6) is a solution 1024x + 289y + 603z = 0 (-2, 5, 1) is a solution 2x + y + z = 2

There is no solution to this set of equations.

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Dependent

1024x + 295y + 605z = 32 (3, 2, - 6) is a solution 1024x + 295y + 605z = 32 (-2, 5, 1) is a solution 2x + y + z = 2

There are an infinite number of solutions to these equations