What is the condition for a unique solution of (ax+by=1) and (2ax+3by=5)?
Answer and explanation
Correct answer: \(ab\neq 0\)
The determinant of the coefficient matrix is \(D=a(3b)-(2a)b=ab\). A pair of linear equations has a unique solution only when \(D\neq 0\). Hence, the required condition is \(ab\neq 0\), meaning that neither \(a\) nor \(b\) can be zero. Conditions such as \(a+b=0\) or \(a=b\) do not by themselves guarantee a unique solution. Exam tip: for a unique solution, check whether \(a_1b_2-a_2b_1\neq0\).
Frequently asked questions
What is the correct answer to this question?
\(ab\neq 0\)
Why is this the correct answer?
The determinant of the coefficient matrix is \(D=a(3b)-(2a)b=ab\). A pair of linear equations has a unique solution only when \(D\neq 0\). Hence, the required condition is \(ab\neq 0\), meaning that neither \(a\) nor \(b\) can be zero. Conditions such as \(a+b=0\) or \(a=b\) do not by themselves guarantee a unique solution. Exam tip: for a unique solution, check whether \(a_1b_2-a_2b_1\neq0\).
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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