When will the pair ((p+2)x+5y=11) and (6x+10y=13) have a unique solution?
Answer and explanation
Correct answer: For every real \(p\) except \(p=1\)
A pair of linear equations has a unique solution when \(\frac{a_1}{a_2}\ne\frac{b_1}{b_2}\). Here, we need \(\frac{p+2}{6}\ne\frac{5}{10}\). Since \(\frac{5}{10}=\frac12\), this gives \(\frac{p+2}{6}\ne\frac12\), so \(p+2\ne3\), or \(p\ne1\). At \(p=1\), the ratios of the coefficients of \(x\) and \(y\) are equal, but the constant ratio \(\frac{11}{13}\) is different; hence there is no solution then. Exam tip: For a unique solution, first compare the ratios of the coefficients of \(x\) and \(y\).
Frequently asked questions
What is the correct answer to this question?
For every real \(p\) except \(p=1\)
Why is this the correct answer?
A pair of linear equations has a unique solution when \(\frac{a_1}{a_2}\ne\frac{b_1}{b_2}\). Here, we need \(\frac{p+2}{6}\ne\frac{5}{10}\). Since \(\frac{5}{10}=\frac12\), this gives \(\frac{p+2}{6}\ne\frac12\), so \(p+2\ne3\), or \(p\ne1\). At \(p=1\), the ratios of the coefficients of \(x\) and \(y\) are equal, but the constant ratio \(\frac{11}{13}\) is different; hence there is no solution then. Exam tip: For a unique solution, first compare the ratios of the coefficients of \(x\) and \(y\).
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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