What is the value of (d) for the equations (6x+dy=54) and (18x+30y=162) to have infinitely many solutions?
Answer and explanation
Correct answer: 10
For two linear equations to have infinitely many solutions, the condition is \(\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}\). Here, \(\frac{6}{18}=\frac{54}{162}=\frac{1}{3}\), so \(\frac{d}{30}=\frac{1}{3}\), giving \(d=10\). If d were 9 or 11, the coefficient ratios would not remain \(1:3\). Exam tip: first compare the ratios of the known coefficients and constants, then use that ratio to find the unknown coefficient.
Frequently asked questions
What is the correct answer to this question?
10
Why is this the correct answer?
For two linear equations to have infinitely many solutions, the condition is \(\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}\). Here, \(\frac{6}{18}=\frac{54}{162}=\frac{1}{3}\), so \(\frac{d}{30}=\frac{1}{3}\), giving \(d=10\). If d were 9 or 11, the coefficient ratios would not remain \(1:3\). Exam tip: first compare the ratios of the known coefficients and constants, then use that ratio to find the unknown coefficient.
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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