What conclusion follows from comparing the ratios of a and b in 6x + 17y = 71 and 11x + 31y = 130?
Answer and explanation
Correct answer: 6/11 ≠ 17/31, so there is one unique solution
Compare the ratios of the corresponding x- and y-coefficients in the equations 6x + 17y = 71 and 11x + 31y = 130. The ratios are 6/11 and 17/31. They are unequal, since 6 × 31 = 186 whereas 17 × 11 = 187. Therefore the two equations represent lines with different slopes. Two lines with different slopes intersect at one and only one point, so the pair is consistent and independent and has a unique solution. It is not necessary to compare the constant ratio after finding the first two ratios unequal. If the first two ratios were equal but the constant ratio differed, the pair would be inconsistent; if all three were equal, it would have infinitely many solutions. Hence option C is correct.
Frequently asked questions
What is the correct answer to this question?
6/11 ≠ 17/31, so there is one unique solution
Why is this the correct answer?
Compare the ratios of the corresponding x- and y-coefficients in the equations 6x + 17y = 71 and 11x + 31y = 130. The ratios are 6/11 and 17/31. They are unequal, since 6 × 31 = 186 whereas 17 × 11 = 187. Therefore the two equations represent lines with different slopes. Two lines with different slopes intersect at one and only one point, so the pair is consistent and independent and has a unique solution. It is not necessary to compare the constant ratio after finding the first two ratios unequal. If the first two ratios were equal but the constant ratio differed, the pair would be inconsistent; if all three were equal, it would have infinitely many solutions. Hence option C is correct.
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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