Dependent System
AlgebraA 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. This happens when the two equations are really the same equation in disguise. Every point that works for one equation works for both.
Example
$x + y = 4$ and $2x + 2y = 8$. The second equation is just the first multiplied by $2$. They are the same line. Infinitely many solutions: $(0,4)$, $(1,3)$, $(2,2)$, etc.
Key Insight
"Dependent" means one equation depends on the other - they carry the same information. Having two equations that say the same thing is like having one equation.
Definition
A dependent system is a consistent system with infinitely many solutions. The equations, when simplified, reduce to the same line. When solved algebraically, a dependent system produces a true statement like $0 = 0$ or $5 = 5$ (an identity), rather than a unique value for the variables.
Example
$4x - 2y = 6$ and $-6x + 3y = -9$. Multiply equation 1 by $-3/2$: $-6x + 3y = -9$. This is exactly equation 2. Dependent. Solutions: any point with $y = 2x - 3$.
Key Insight
The solution set of a dependent 2-variable system is described parametrically: let $x = t$, then $y = 2t - 3$ (for example). The parameter $t$ can be any real number, giving infinitely many solutions.
Definition
A dependent system is a consistent system where $\text{rank}(A) = \text{rank}([A|b]) < n$ (number of unknowns). The solution space is an affine subspace of dimension $n - \text{rank}(A)$. For two equations in two unknowns with rank $1$, the solution space is a $1$-dimensional affine subspace (a line). The general solution is $x = x_p + x_h$, where $x_p$ is any particular solution and $x_h$ is a vector in the null space of $A$.
Example
$A = [[2,-1],[4,-2]]$, $b = [[3],[6]]$. $\text{rank} = 1$. Null space of $A$ is spanned by $(1, 2)$. Particular solution: $x_p = (3/2, 0)$. General solution: $(3/2 + t, 2t)$ for any $t$ in $\mathbb{R}$.
Key Insight
Dependent systems arise whenever there is redundancy in the constraints. In data science, this corresponds to multicollinearity, where predictor variables are linearly related, making unique parameter estimates impossible.