Which condition is correct for (6x+ky=24) and (3x+5y=17) to have a unique solution?
Answer and explanation
Correct answer: \(k\ne 10\)
Two linear equations \(a_1x+b_1y=c_1\) and \(a_2x+b_2y=c_2\) have a unique solution when \(\frac{a_1}{a_2}\ne\frac{b_1}{b_2}\). Here, \(\frac{6}{3}=2\) and the other coefficient ratio is \(\frac{k}{5}\). Thus, for a unique solution, \(2\ne\frac{k}{5}\), which gives \(k\ne10\). If \(k=10\), the coefficient ratios are equal, but \(\frac{24}{17}\ne2\), so the equations have no solution. Exam tip: compare the ratios of the coefficients of \(x\) and \(y\) first.
Frequently asked questions
What is the correct answer to this question?
\(k\ne 10\)
Why is this the correct answer?
Two linear equations \(a_1x+b_1y=c_1\) and \(a_2x+b_2y=c_2\) have a unique solution when \(\frac{a_1}{a_2}\ne\frac{b_1}{b_2}\). Here, \(\frac{6}{3}=2\) and the other coefficient ratio is \(\frac{k}{5}\). Thus, for a unique solution, \(2\ne\frac{k}{5}\), which gives \(k\ne10\). If \(k=10\), the coefficient ratios are equal, but \(\frac{24}{17}\ne2\), so the equations have no solution. Exam tip: compare the ratios of the coefficients of \(x\) and \(y\) first.
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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