What is the most suitable conclusion by observing the equations (9x+14y=71) and (27x+42y=213)?
Answer and explanation
Correct answer: There are infinitely many solutions
Every term in the second equation is three times the corresponding term in the first: 27=3×9, 42=3×14, and 213=3×71. Thus, both equations represent the same line, so every point on that line satisfies both equations. Therefore, the pair has infinitely many solutions. A no-solution or distinct-parallel-lines conclusion would require the coefficients of x and y to be proportional but the constants not to be proportional; here all three ratios are equal. Exam tip: If a₁/a₂ = b₁/b₂ = c₁/c₂, the pair has infinitely many solutions.
Frequently asked questions
What is the correct answer to this question?
There are infinitely many solutions
Why is this the correct answer?
Every term in the second equation is three times the corresponding term in the first: 27=3×9, 42=3×14, and 213=3×71. Thus, both equations represent the same line, so every point on that line satisfies both equations. Therefore, the pair has infinitely many solutions. A no-solution or distinct-parallel-lines conclusion would require the coefficients of x and y to be proportional but the constants not to be proportional; here all three ratios are equal. Exam tip: If a₁/a₂ = b₁/b₂ = c₁/c₂, the pair has infinitely many solutions.
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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