For what value of k does the system of equations x − ky = 2 and 3x + 2y = −5…
2023
For what value of k does the system of equations x − ky = 2 and 3x + 2y = −5 have a unique solution?
Answer: D. More than one of the above — ConceptFor a 2 × 2 linear system a1x + b1y = c1 and a2x + b2y = c2, a unique solution exists exactly when the determinant Δ = a1b2 − a2b1 is non-zero.…
- A.
−2/3
- B.
1
- C.
2
- D.
More than one of the above
- E.
None of the above
Attempted by 2 students.
Show answer & explanation
Correct answer: D
Concept
For a 2 × 2 linear system a1x + b1y = c1 and a2x + b2y = c2, a unique solution exists exactly when the determinant Δ = a1b2 − a2b1 is non-zero.
Geometrically, a non-zero determinant means the coefficient rows are not proportional, so the two lines intersect once.
Application
Here a1 = 1, b1 = −k, a2 = 3, and b2 = 2.
Compute Δ = (1)(2) − (3)(−k) = 2 + 3k.
A unique solution requires 2 + 3k ≠ 0, so k ≠ −2/3.
Among the listed numerical values, k = 1 and k = 2 both satisfy k ≠ −2/3. Therefore, more than one listed value gives a unique solution.
Cross-check
For k = 1, Δ = 5; for k = 2, Δ = 8; both are non-zero. For k = −2/3, Δ = 0. Therefore, exactly two listed numerical values satisfy the uniqueness condition.