When will ((p-2)x+9y=4) and (5x+15y=12) have a unique solution?
Answer and explanation
Correct answer: \(p\neq 5\)
Two linear equations \(a_1x+b_1y=c_1\) and \(a_2x+b_2y=c_2\) have a unique solution when \(\frac{a_1}{a_2}\neq\frac{b_1}{b_2}\). Here, we need \(\frac{p-2}{5}\neq\frac{9}{15}=\frac{3}{5}\). The ratios become equal only when \(p-2=3\), i.e. \(p=5\). Therefore, the pair has a unique solution for \(p\neq 5\). At \(p=2\), the ratios are still unequal, so it does not prevent a unique 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?
\(p\neq 5\)
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 \(\frac{a_1}{a_2}\neq\frac{b_1}{b_2}\). Here, we need \(\frac{p-2}{5}\neq\frac{9}{15}=\frac{3}{5}\). The ratios become equal only when \(p-2=3\), i.e. \(p=5\). Therefore, the pair has a unique solution for \(p\neq 5\). At \(p=2\), the ratios are still unequal, so it does not prevent a unique 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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