Which of the following conditions gives a unique solution for a pair of linear equations?
Answer and explanation
Correct answer: a₁/a₂ ≠ b₁/b₂
For a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0, a unique solution occurs when the two corresponding lines intersect at exactly one point. Algebraically, this happens when a₁/a₂ ≠ b₁/b₂. The unequal ratios mean the lines have different slopes and cannot be parallel or coincident, so they meet once. Option A describes coincident lines and gives infinitely many solutions. Option B describes parallel distinct lines and gives no solution. Option D is incomplete and is not a valid general condition. Hence option C is the only correct condition.
Frequently asked questions
What is the correct answer to this question?
a₁/a₂ ≠ b₁/b₂
Why is this the correct answer?
For a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0, a unique solution occurs when the two corresponding lines intersect at exactly one point. Algebraically, this happens when a₁/a₂ ≠ b₁/b₂. The unequal ratios mean the lines have different slopes and cannot be parallel or coincident, so they meet once. Option A describes coincident lines and gives infinitely many solutions. Option B describes parallel distinct lines and gives no solution. Option D is incomplete and is not a valid general condition. Hence option C is the only correct condition.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Pair of Linear Equations in Two Variables. Topic: Conditions for solvability.
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