Dependent System
A dependent system of equations has infinitely many solutions because the equations represent the same line and every point on that line is a solution.
Definition
A dependent system has infinitely many solutions because its equations, once simplified, are really the same equation in disguise, reducing to the same line; every point that satisfies one equation satisfies both. Solved algebraically, a dependent system produces a true statement like $0 = 0$ rather than a unique value, and its two-variable solution set can be described parametrically, letting $x = t$ gives $y$ as some expression in $t$, for any real number $t$. Formally, a dependent system is consistent with $\text{rank}(A) = \text{rank}([A|b]) < n$, so the solution space is an affine subspace of dimension $n - \text{rank}(A)$; the general solution is $x = x_p + x_h$, where $x_p$ is any particular solution and $x_h$ lies in the null space of $A$.
Example
$x + y = 4$ and $2x + 2y = 8$ are the same line (the second is just the first doubled), giving infinitely many solutions like $(0,4)$, $(1,3)$, $(2,2)$. Multiplying $4x - 2y = 6$ by $-3/2$ gives exactly $-6x + 3y = -9$, showing the two equations describe the same line, with solutions any point where $y = 2x - 3$. For $A = [[2,-1],[4,-2]]$, $b = [[3],[6]]$, $\text{rank}(A) = 1$; the null space is spanned by $(1, 2)$, and with particular solution $x_p = (3/2, 0)$, the general solution is $(3/2 + t, 2t)$ for any $t$.
Key Insight
"Dependent" means one equation depends on the other, they carry the same information, so having two equations that say the same thing is like having only one. This redundancy shows up in data science as multicollinearity, where predictor variables are linearly related, making unique parameter estimates impossible.