If a₁/a₂ = b₁/b₂ = c₁/c₂, how many solutions will the pair of linear equations have?
Answer and explanation
Correct answer: Infinitely many solutions
For the pair a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0, the condition a₁/a₂ = b₁/b₂ = c₁/c₂ means that every coefficient, including the constant term, is in the same ratio. Consequently, one equation is a scalar multiple of the other, so both equations represent the same line rather than two distinct lines. Every point on that common line satisfies both equations. Therefore, the pair is dependent and consistent and has infinitely many solutions. A unique solution would require unequal coefficient ratios, while no solution occurs when the first two ratios agree but the constant-term ratio differs.
Frequently asked questions
What is the correct answer to this question?
Infinitely many solutions
Why is this the correct answer?
For the pair a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0, the condition a₁/a₂ = b₁/b₂ = c₁/c₂ means that every coefficient, including the constant term, is in the same ratio. Consequently, one equation is a scalar multiple of the other, so both equations represent the same line rather than two distinct lines. Every point on that common line satisfies both equations. Therefore, the pair is dependent and consistent and has infinitely many solutions. A unique solution would require unequal coefficient ratios, while no solution occurs when the first two ratios agree but the constant-term ratio differs.
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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