Question

A pair of linear equations \( a_1 x + b_1 y + c_1 = 0 \) and \( a_2 x + b_2 y + c_2 = 0 \) is such that \( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \), then they are:

• A. consistent
• B. inconsistent
• C. dependent and consistent
• D. None of these

Hint:

For two linear equations, if the ratio of the coefficients of \( x \) and \( y \) is unequal, the system is consistent, and the equations have a unique solution.

Answer 1

The Correct Option is A

Step 1: Understand the condition for the system of equations
The given pair of linear equations is: \[ a_1 x + b_1 y + c_1 = 0 \quad \text{and} \quad a_2 x + b_2 y + c_2 = 0 \] The condition is \( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \). Step 2: Determine the type of system of equations
For a system of two linear equations, if the coefficients of \( x \) and \( y \) (i.e., \( a_1, b_1, a_2, b_2 \)) are such that \( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \), the system is consistent. This means the lines represented by the equations will intersect at a unique point, and hence, the system will have a unique solution. Step 3: Conclusion
Since \( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \), the system is consistent, meaning the equations will have a unique solution. Therefore, the correct answer is option \( (1) \).