1. Let's state the problem: Determine when a system of linear equations has one solution, no solution, or infinitely many solutions. 2. Consider a system of two linear equations in two variables: $$a_1x + b_1y = c_1$$ $$a_2x + b_2y = c_2$$ 3. The key is to analyze the ratios of the coefficients: - If $$\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$$, the lines intersect at exactly one point, so there is **one unique solution**. - If $$\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$$, the lines are parallel but not the same line, so there is **no solution**. - If $$\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$$, the two equations represent the same line, so there are **infinitely many solutions**. 4. In summary: - One solution: lines intersect (different slopes). - No solution: lines parallel and distinct. - Infinite solutions: lines coincide. This rule applies generally to linear systems and helps quickly determine the nature of solutions.