If 10x − 5y = 30 and 2x − y = t are inconsistent, what is the correct condition on t?
Answer and explanation
Correct answer: t ≠ 6
For a pair of linear equations a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0, the equations are inconsistent when a₁/a₂ = b₁/b₂ but c₁/c₂ is different. Here, 10x − 5y = 30 can be compared with 2x − y = t. The coefficient ratios are 10/2 = 5 and (−5)/(−1) = 5, so the two lines have the same slope. They will be distinct parallel lines, and hence have no solution, only when 30/t ≠ 5, which is equivalent to t ≠ 6. If t = 6, the second equation multiplied by 5 becomes the first, giving infinitely many solutions rather than inconsistency. Therefore option B is correct; the other numerical values do not express the required condition.
Frequently asked questions
What is the correct answer to this question?
t ≠ 6
Why is this the correct answer?
For a pair of linear equations a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0, the equations are inconsistent when a₁/a₂ = b₁/b₂ but c₁/c₂ is different. Here, 10x − 5y = 30 can be compared with 2x − y = t. The coefficient ratios are 10/2 = 5 and (−5)/(−1) = 5, so the two lines have the same slope. They will be distinct parallel lines, and hence have no solution, only when 30/t ≠ 5, which is equivalent to t ≠ 6. If t = 6, the second equation multiplied by 5 becomes the first, giving infinitely many solutions rather than inconsistency. Therefore option B is correct; the other numerical values do not express the required condition.
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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