Consider the system of linear equations:
$\begin{aligned} x + 2y + 3z &= 4 \\ 2x + 5y + 7z &= 9 \\ 3x + 7y + c z &= 10 \end{aligned}$
The system has NO solution when the parameter $c$ equals ________ (answer in integer).
Answer Key:
10
Step-by-Step Solution & Explanation
**Method 1: Gaussian Elimination on Augmented Matrix**
Form the augmented matrix $[A|B]$:
$\begin{pmatrix} 1 & 2 & 3 & | & 4 \\ 2 & 5 & 7 & | & 9 \\ 3 & 7 & c & | & 10 \end{pmatrix}$
Perform row operations:
$R_2 \to R_2 - 2R_1$:
$\begin{pmatrix} 0 & 1 & 1 & | & 1 \end{pmatrix}$
$R_3 \to R_3 - 3R_1$:
$\begin{pmatrix} 0 & 1 & c - 9 & | & -2 \end{pmatrix}$
$R_3 \to R_3 - R_2$:
$\begin{pmatrix} 0 & 0 & c - 10 & | & -3 \end{pmatrix}$
For the system to have NO solution (inconsistent):
$\operatorname{rank}(A) < \operatorname{rank}([A|B])$
This occurs when the coefficient entry in the last row vanishes while the augmented entry is non-zero:
$c - 10 = 0 \implies c = 10$
When $c = 10$, $\operatorname{rank}(A) = 2$ and $\operatorname{rank}([A|B]) = 3$, so no solution exists.