Question

The pair of linear equations \(a_1x+b_1y+c_1=0\) and \(a_2x+b_2y+c_2=0\) has a unique solution, then

• A. \(\frac {a_1}{a_2} ≠ \frac {b_1}{b_2}\)
• B. \(\frac {a_1}{a_2} = \frac {b_1}{b_2} ≠ \frac {c_1}{c_2}\)
• C. \(\frac {a_1}{a_2} = \frac {b_1}{b_2} = \frac {c_1}{c_2}\)
• D. \(\frac {a_1}{a_2} = \frac {b_1}{b_2}\)

Answer 1

The Correct Option is A

To determine the condition for the pair of linear equations to have a unique solution, let's analyze the given equations: \[ \begin{cases} a_1x + b_1y + c_1 = 0 \\ a_2x + b_2y + c_2 = 0 \end{cases} \] 

Key Concept: A system of linear equations has a unique solution if and only if the lines represented by the equations intersect at exactly one point. This occurs when the lines are not parallel, which mathematically translates to the ratios of their coefficients not being equal. Condition for a Unique Solution: For the system to have a unique solution, the following must hold: \[ \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \] This ensures that the lines are not parallel and thus intersect at a single point. 

Final Answer: The correct condition for a unique solution is: \[ \boxed{\frac{a_1}{a_2} \neq \frac{b_1}{b_2}} \] Thus, the correct option is (1).