If two equations satisfy a₁/a₂ = b₁/b₂ and this common ratio is also equal to c₁/c₂, what happens?
Answer and explanation
Correct answer: There are infinitely many solutions
For two linear equations a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0, the condition a₁/a₂ = b₁/b₂ = c₁/c₂ means that all coefficients, including the constants, are in the same proportion. One equation is therefore a scalar multiple of the other, so both represent the same line. Since every point on that line satisfies both equations, the pair has infinitely many solutions. Unequal constant ratios would instead indicate distinct parallel lines.
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What is the correct answer to this question?
There are infinitely many solutions
Why is this the correct answer?
For two linear equations a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0, the condition a₁/a₂ = b₁/b₂ = c₁/c₂ means that all coefficients, including the constants, are in the same proportion. One equation is therefore a scalar multiple of the other, so both represent the same line. Since every point on that line satisfies both equations, the pair has infinitely many solutions. Unequal constant ratios would instead indicate distinct parallel lines.
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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