Consider the system of equations $ax + by = 0$ and $cx + dy = 0$ where $a,b,c,d \in \{ 1,2\}$. Then

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Consider the system of equations $ax + by = 0$ and $cx + dy = 0$ where $a,b,c,d \in \{ 1,2\}$. Then the probability that the system of equations has

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Sol. [Key Concept: Unique/non-trivial solution $\Leftrightarrow \ ad - bc 
eq 0$; count favorable $(a,b,c,d)$.]$ad - bc 
eq 0$ for unique soln. ad bc $$(a,d,b,c)$$ 1 2 or $4$ $$(1,1,1,2),(1,1,2,1),(1,1,2,2)$$ 2 1 or $4$ $$(1,2,1,1),(2,1,1,1),(1,2,2,2),(2,1,2,2)$$ 3 1 or $2$ $$(2,2,1,1),(2,2,1,2),(2,2,2,1)$$ $$P = \frac{10}{16} = \frac{5}{8}$$∴ for non-trivial soln.$$P = \frac{3}{8}$$

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ad	bc	$$(a,d,b,c)$$
1	2 or $4$	$$(1,1,1,2),(1,1,2,1),(1,1,2,2)$$
2	1 or $4$	$$(1,2,1,1),(2,1,1,1),(1,2,2,2),(2,1,2,2)$$
3	1 or $2$	$$(2,2,1,1),(2,2,1,2),(2,2,2,1)$$