Explain the conditions for a pair of linear equations $a1x + b1y + c1 = 0$ and $a2x + b2y + c2 = 0$ to have: (i) a unique solution, (ii) infinitely many solutions, and (iii) no solution. What do these conditions represent graphically?

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Consider the pair of linear equations in two variables: $$a1x + b1y + c1 = 0$$ $$a2x + b2y + c2 = 0$$ Unique Solution: A pair of linear equations has a unique solution if: $$\frac{a1}{a2} 
eq \frac{b1}{b2}$$ Graphical Representation: Graphically, the two lines represented by these equations intersect at exactly one point. The coordinates of this point give the unique solution for $x$ and $y$. Such a pair of linear equations is called a consistent system. Infinitely Many Solutions: A pair of linear equations has infinitely many solutions if: $$\frac{a1}{a2} = \frac{b1}{b2} = \frac{c1}{c2}$$ Graphical Representation: Graphically, the two lines coincide (overlap completely) with each other. Every point on the line is a common solution to both equations. Such a system is called a dependent and consistent system. No Solution: A pair of linear equations has no solution if: $$\frac{a1}{a2} = \frac{b1}{b2} 
eq \frac{c1}{c2}$$ Graphical Representation: Graphically, the two lines are parallel to each other and will never intersect at any point. Therefore, there is no common solution for the variables. Such a pair of linear equations is called an inconsistent system.