What is the relation among all three ratios in the equations (5x+6y=32) and (15x+18y=96)?
Answer and explanation
Correct answer: All three ratios are equal
Here, \(a_1/a_2=5/15=1/3\), \(b_1/b_2=6/18=1/3\), and \(c_1/c_2=32/96=1/3\). Thus, all three ratios are equal. Therefore, the two equations represent the same line and have infinitely many solutions. Option B would apply if only the first two ratios were equal, but here the constant-term ratio is also equal. Exam tip: if \(a_1/a_2=b_1/b_2=c_1/c_2\), the lines are coincident.
Frequently asked questions
What is the correct answer to this question?
All three ratios are equal
Why is this the correct answer?
Here, \(a_1/a_2=5/15=1/3\), \(b_1/b_2=6/18=1/3\), and \(c_1/c_2=32/96=1/3\). Thus, all three ratios are equal. Therefore, the two equations represent the same line and have infinitely many solutions. Option B would apply if only the first two ratios were equal, but here the constant-term ratio is also equal. Exam tip: if \(a_1/a_2=b_1/b_2=c_1/c_2\), the lines are coincident.
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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