What is the solution status of 3x + 5y = 14 and 6x + 11y = 30?
Answer and explanation
Correct answer: One unique solution
The governing concept is the ratio test for a pair of linear equations, a x + b y + c = 0 and a₁x + b₁y + c₁ = 0. Here a/a₁ = 3/6 = 1/2, but b/b₁ = 5/11, and these ratios are unequal. Therefore the two lines have different slopes and intersect at exactly one point. A pair with one intersection has one unique solution, so option A is correct. Infinitely many solutions would require all three corresponding ratios to be equal, while no solution would require a/a₁ = b/b₁ but not c/c₁. Since the coefficient ratios already differ, neither of those cases applies.
Frequently asked questions
What is the correct answer to this question?
One unique solution
Why is this the correct answer?
The governing concept is the ratio test for a pair of linear equations, a x + b y + c = 0 and a₁x + b₁y + c₁ = 0. Here a/a₁ = 3/6 = 1/2, but b/b₁ = 5/11, and these ratios are unequal. Therefore the two lines have different slopes and intersect at exactly one point. A pair with one intersection has one unique solution, so option A is correct. Infinitely many solutions would require all three corresponding ratios to be equal, while no solution would require a/a₁ = b/b₁ but not c/c₁. Since the coefficient ratios already differ, neither of those cases applies.
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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