This problem involves analyzing a system of linear equations using determinants, a fundamental concept in Linear Algebra. We will use Cramer's Rule to determine the nature of the solutions (unique, infinitely many, or no solution) based on the value of the parameter 'a'.

1. Key Concepts: Cramer's Rule for Systems of Linear Equations

Consider a system of $n$ linear equations in $n$ variables: $a_{11}x_1 + a_{12}x_2 + \dots + a_{1n}x_n = b_1$ $a_{21}x_1 + a_{22}x_2 + \dots + a_{2n}x_n = b_2$ $\vdots$