To determine the conditions for a linear system to be consistent, we follow these steps:
Key Concept
A linear system (Ax = B) is consistent if and only if the rank of the augmented matrix ([A|B]) equals the rank of matrix (A). This means no row in the reduced row-echelon form of ([A|B]) should be of the form ([0\ 0\ ...\ 0\ |\ k]) where (k \neq 0).
General Approach
- Form the Augmented Matrix: Combine the coefficient matrix (A) and the constant vector (B) into ([A|B]).
- Row Reduction: Perform Gaussian elimination to reduce ([A|B]) to row-echelon form.
- Check for Inconsistency: Identify rows with all zero coefficients for variables. For consistency, their constant terms must be zero.
Example (Common Problem)
Suppose the system is:
[
\begin{align}
x + 2y - 3z &= a \
2x + 3y + 3z &= b \
5x + 9y - 6z &= c \
\end{align}
]
Step 1: Augmented matrix:
[
\begin{bmatrix}
1 & 2 & -3 & | & a \
2 & 3 & 3 & | & b \
5 & 9 & -6 & | & c \
\end{bmatrix}
]
Step 2: Row operations:
- (R_2 = R_2 - 2R_1): (\begin{bmatrix}0 & -1 & 9 & | & b - 2a\end{bmatrix})
- (R_3 = R_3 -5R_1): (\begin{bmatrix}0 & -1 & 9 & | & c -5a\end{bmatrix})
- (R_3 = R_3 - R_2): (\begin{bmatrix}0 & 0 & 0 & | & (c -5a) - (b -2a)\end{bmatrix} = \begin{bmatrix}0 & 0 &0 & | & c -b -3a\end{bmatrix})
Step 3: For consistency:
(c - b -3a = 0 \implies \boxed{3a + b = c})
Final Answer (for this example): (\boxed{3a + b = c})
(Adjust based on the actual system, but the method remains the same.)
If the original system differs, apply the same steps to derive the condition. For most standard problems, the result is a linear relation between (a, b, c).
Example Output: (\boxed{a + b = c}) (if the system leads to that condition) or similar.
Assuming the common problem, the answer is (\boxed{3a + b = c}).
(\boxed{3a + b = c})
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