Discussion
The system of linear equations is consistent :
\[ \begin{pmatrix} 2 & 0 & 6 \\ 2 & 2 & -4 \end{pmatrix} \begin{pmatrix} x_{1} \\ x_{2} \\ x_{3} \end{pmatrix} = \begin{pmatrix} b_{1} \\ b_{2} \end{pmatrix}. \]
Answer: D
The system is consistent for all values of b1 and b2 and has infinitely many solutions.
The coefficient matrix has rows (2, 0, 6) and (2, 2, −4), which are not multiples of each other, so its rank is 2. With only two rows, the augmented matrix also has rank 2 for any right-hand side, so the system is always consistent.
There are 3 unknowns and rank 2, so one variable is free:
\[ \text{free variables} = 3 - 2 = 1 \]
A free variable gives infinitely many solutions, never a unique one. Nothing special happens at \(b_{1} = 4,\ b_{2} = -12\), so the options restricting the right-hand side are wrong.
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