Consider the following linear system. x + 2y – 3z = a 2x + 3y + 3z = b 5x + 9y – 6z = c This system is consistent if a, b and c satisfy the equation A. 7a – b – c = 0 B. 3a + b – c = 0 C. 3a – b + c = 0 D. 7a – b + c = 0

The correct answer is: A. 7a – b – c = 0

A system of linear equations is consistent if it has at least one solution. A system of linear equations is said to be dependent if it has infinitely many solutions, and independent if it has exactly one solution.

To determine whether a system of linear equations is consistent, we can use the following theorem:

A system of linear equations is consistent if and only if the determinant of the coefficient matrix is not equal to zero.

In this case, the coefficient matrix is:

[1 2 -3]
[2 3 3]
[5 9 -6]

The determinant of this matrix is:

7a – b – c

Therefore, the system of linear equations is consistent if and only if 7a – b – c is not equal to zero.

Option A: 7a – b – c = 0

This option satisfies the condition that 7a – b – c is not equal to zero. Therefore, the system of linear equations is consistent.

Option B: 3a + b – c = 0

This option does not satisfy the condition that 7a – b – c is not equal to zero. Therefore, the system of linear equations is not consistent.

Option C: 3a – b + c = 0

This option does not satisfy the condition that 7a – b – c is not equal to zero. Therefore, the system of linear equations is not consistent.

Option D: 7a – b + c = 0

This option does not satisfy the condition that 7a – b – c is not equal to zero. Therefore, the system of linear equations is not consistent.