Let $a, \lambda, \mu \in \mathbb{R}$. Consider the system of linear equations $ax + 2y = \lambda$ $3x - 2y = \mu$ Which of the following statement(s) is (are) correct?

$ax+2y=\lambda$

$3x-2y=\mu$

$a,\ \lambda,\ \mu \in \mathbb{R}$

$\text{To find: which statements about consistency are correct}$

A= a 2 3 -2 , D= a 2 3 -2

$D=a(-2)-(2)(3)=-2a-6=-2(a+3)$

$\text{Plan: } D\neq 0 \Rightarrow \text{unique};\quad D=0 \Rightarrow \text{check consistency}$

$a\neq -3:\ D\neq 0 \Rightarrow \text{unique solution for all } \lambda,\mu \;\Rightarrow\; (2)\ \text{TRUE}$

$a=-3:\ -3x+2y=\lambda,\qquad 3x-2y=\mu$

$3x-2y=\mu \;\xrightarrow{\times(-1)}\; -3x+2y=-\mu$

$\text{Consistent} \iff \lambda=-\mu \iff \lambda+\mu=0$

$\lambda+\mu=0 \Rightarrow \text{coincident lines} \Rightarrow \infty \text{ solutions} \;\Rightarrow\; (3)\ \text{TRUE}$

$\lambda+\mu\neq 0 \Rightarrow \text{parallel lines} \Rightarrow \text{no solution} \;\Rightarrow\; (4)\ \text{TRUE}$

$(1)\ \text{false: } a=-3 \text{ gives } \infty \text{ only if } \lambda+\mu=0$

$\text{Correct: } (2),\ (3),\ (4)$