What is the condition for a unique solution of (ax+2y=3) and (4x+ay=9)?
Answer and explanation
Correct answer: \(a^2\neq 8\)
A pair of linear equations has a unique solution when the determinant of its coefficient matrix is non-zero. Here, the determinant is \(a\cdot a-4\cdot2=a^2-8\). Therefore, for a unique solution, \(a^2-8\neq0\), i.e. \(a^2\neq8\). The condition \(a\neq2\) is not sufficient because the determinant is also zero at \(a=-2\sqrt2\). Exam tip: for a unique solution, check that \(a_1b_2-a_2b_1\neq0\).
Frequently asked questions
What is the correct answer to this question?
\(a^2\neq 8\)
Why is this the correct answer?
A pair of linear equations has a unique solution when the determinant of its coefficient matrix is non-zero. Here, the determinant is \(a\cdot a-4\cdot2=a^2-8\). Therefore, for a unique solution, \(a^2-8\neq0\), i.e. \(a^2\neq8\). The condition \(a\neq2\) is not sufficient because the determinant is also zero at \(a=-2\sqrt2\). Exam tip: for a unique solution, check that \(a_1b_2-a_2b_1\neq0\).
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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