Question:
Consider the following system of equations :
\(x + 2y - 3z = a\)
\(2x + 6y - 11z = b\)
\(x - 2y + 7z = c\),
where \(a\), \(b\) and \(c\) are real constants. Then the system of equations :
Consider the following system of equations :
\(x + 2y - 3z = a\)
\(2x + 6y - 11z = b\)
\(x - 2y + 7z = c\),
where \(a\), \(b\) and \(c\) are real constants. Then the system of equations :
\(x + 2y - 3z = a\)
\(2x + 6y - 11z = b\)
\(x - 2y + 7z = c\),
where \(a\), \(b\) and \(c\) are real constants. Then the system of equations :
Show Hint
The coefficient determinant is 0, so find the linear relation between the left sides; consistency requires the same relation on a, b, c.
Updated On: Nov 4, 2026
- has no solution for all \(a\), \(b\) and \(c\)
- has a unique solution when \(5a = 2b + c\)
- has infinite number of solutions when \(5a = 2b + c\)
- has a unique solution for all \(a\), \(b\) and \(c\)
Show Solution
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The Correct Option is C
Solution and Explanation
Coefficient determinant: \(\begin{vmatrix}1&2&-3\\2&6&-11\\1&-2&7\end{vmatrix} = 1(42-22) - 2(14+11) - 3(-4-6) = 20 - 50 + 30 = 0\).
So the system has either no solution or infinitely many.
Find \(\alpha, \beta\) with \(R_3 = \alpha R_1 + \beta R_2\): \(\alpha + 2\beta = 1\), \(2\alpha + 6\beta = -2\) give \(\beta = -2\), \(\alpha = 5\); check on the z-column: \(-15 + 22 = 7\), true.
So \(c = 5a - 2b\), i.e. \(5a = 2b + c\) is needed for consistency.
When \(5a = 2b + c\), the system is consistent with rank 2 < 3 unknowns, hence infinitely many solutions; otherwise no solution.
Answer: (3) has infinite number of solutions when \(5a = 2b + c\)
So the system has either no solution or infinitely many.
Find \(\alpha, \beta\) with \(R_3 = \alpha R_1 + \beta R_2\): \(\alpha + 2\beta = 1\), \(2\alpha + 6\beta = -2\) give \(\beta = -2\), \(\alpha = 5\); check on the z-column: \(-15 + 22 = 7\), true.
So \(c = 5a - 2b\), i.e. \(5a = 2b + c\) is needed for consistency.
When \(5a = 2b + c\), the system is consistent with rank 2 < 3 unknowns, hence infinitely many solutions; otherwise no solution.
Answer: (3) has infinite number of solutions when \(5a = 2b + c\)
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