Question

Consider the following system of linear equations: 
\[ \begin{cases} x + y + 5z = 3, \\ x + 2y + mz = 5, \\ x + 2y + 4z = k. \end{cases} \] 

The system is consistent if 
 

• A. \( m \neq 4 \)
• B. \( k \neq 5 \)
• C. \( m = 4 \)
• D. \( k = 5 \)

Hint:

When analyzing system consistency, use Gaussian elimination and check when a zero row yields a contradiction like \( 0 = c \).

Answer 1

The Correct Option is A, D

Step 1: Write the augmented matrix. 
\[ \begin{bmatrix} 1 & 1 & 5 & | & 3 \\ 1 & 2 & m & | & 5 \\ 1 & 2 & 4 & | & k \end{bmatrix} \] Subtract the first row from the others: \[ \begin{bmatrix} 1 & 1 & 5 & | & 3 \\ 0 & 1 & m - 5 & | & 2 \\ 0 & 1 & -1 & | & k - 3 \end{bmatrix}. \] Subtract the second row from the third: \[ \begin{bmatrix} 1 & 1 & 5 & | & 3 \\ 0 & 1 & m - 5 & | & 2 \\ 0 & 0 & -m + 4 & | & k - 5 \end{bmatrix}. \] 

Step 2: Condition for consistency.
For the system to be consistent, the last equation must not become contradictory. If \( m \neq 4 \), the third equation gives a valid value for \( z \). If \( m = 4 \), the coefficient of \( z \) vanishes, and we must have \( k - 5 = 0 \Rightarrow k = 5 \) for consistency. Thus, the system is consistent for all \( m \neq 4 \) and also for \( m = 4, k = 5 \). 

Step 3: Simplify conclusion.
Hence, the general condition ensuring consistency is \( m \neq 4 \), except for one special case. 
 

Final Answer: \[ \boxed{m \neq 4} \]