Question

If \(\frac{a_1}{a_2} \neq \frac{b_1}{b_2}\), where \(a_1x + b_1y + c_1 = 0\) and \(a_2x + b_2y + c_2 = 0\) are two linear equations, then the equations

• A. have a unique solution
• B. have infinitely many solutions
• C. have finite solutions
• D. have no solution

Hint:

For a system of linear equations, if the coefficients of \(x\) and \(y\) are in proportion, the system may have infinitely many solutions or no solution, but if they are not proportional, the system has exactly one solution.

Answer 1

The Correct Option is A

For two linear equations in the form \(a_1x + b_1y + c_1 = 0\) and \(a_2x + b_2y + c_2 = 0\), if \(\frac{a_1}{a_2} \neq \frac{b_1}{b_2}\), then the two lines represented by the equations are not parallel and will intersect at exactly one point. This means the system has a unique solution. Thus, the correct answer is option (1).