For the prices of two items, the equations 2x + 7y = 160 and 5x + 16y = 370 are formed. What is the solution status?
Answer and explanation
Correct answer: One unique solution
The solution status of two linear equations can be determined by comparing a₁/a₂ and b₁/b₂. For 2x + 7y = 160 and 5x + 16y = 370, the first coefficient ratio is 2/5, while the second is 7/16. These are unequal because 2 × 16 = 32 whereas 7 × 5 = 35. Thus the two equations represent lines with different slopes. Lines with different slopes intersect at exactly one point, so the pair is consistent and independent and has one unique solution. The constant terms are not needed once the first two ratios are unequal. “No solution” would require equal coefficient ratios but a different constant ratio; infinitely many solutions would require all three ratios to be equal. Therefore option A is correct.
Frequently asked questions
What is the correct answer to this question?
One unique solution
Why is this the correct answer?
The solution status of two linear equations can be determined by comparing a₁/a₂ and b₁/b₂. For 2x + 7y = 160 and 5x + 16y = 370, the first coefficient ratio is 2/5, while the second is 7/16. These are unequal because 2 × 16 = 32 whereas 7 × 5 = 35. Thus the two equations represent lines with different slopes. Lines with different slopes intersect at exactly one point, so the pair is consistent and independent and has one unique solution. The constant terms are not needed once the first two ratios are unequal. “No solution” would require equal coefficient ratios but a different constant ratio; infinitely many solutions would require all three ratios to be equal. Therefore option A 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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