Which of the following conditions gives a unique solution for a pair of linear equations?
Answer and explanation
Correct answer: a₁/a₂ ≠ b₁/b₂
Consider the pair a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0. A unique solution means that the corresponding lines intersect at exactly one point. This occurs when their slopes are different, which is represented algebraically by a₁/a₂ ≠ b₁/b₂. Since the coefficient ratios are unequal, the lines cannot be parallel or coincident, so one common ordered pair satisfies both equations. Option A, where all three ratios are equal, represents coincident lines and infinitely many solutions. Option B represents distinct parallel lines and no solution. Option D is incomplete and is not a general criterion. Therefore 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?
Consider the pair a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0. A unique solution means that the corresponding lines intersect at exactly one point. This occurs when their slopes are different, which is represented algebraically by a₁/a₂ ≠ b₁/b₂. Since the coefficient ratios are unequal, the lines cannot be parallel or coincident, so one common ordered pair satisfies both equations. Option A, where all three ratios are equal, represents coincident lines and infinitely many solutions. Option B represents distinct parallel lines and no solution. Option D is incomplete and is not a general criterion. Therefore 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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