Independent System
AlgebraAn independent system of equations has exactly one solution, corresponding to two lines that intersect at precisely one point.
Definition
An independent system has exactly one solution. The two equations each contain unique information, and together they pinpoint a single answer.
Example
$y = x + 2$ and $y = 3x - 4$. Setting equal: $x + 2 = 3x - 4$, so $6 = 2x$, $x = 3$, $y = 5$. One solution: $(3, 5)$.
Key Insight
"Independent" means the equations give different information. Together they narrow down the answer to exactly one point.
Definition
A consistent independent system has exactly one solution. The lines are not parallel and not the same, so they cross at exactly one point. Algebraically, solving the system yields unique values for all variables, not a contradiction or an identity.
Example
$2x + y = 7$ and $x - y = 2$. Add: $3x = 9$, $x = 3$, $y = 1$. The solution $(3, 1)$ is the unique intersection point. Neither equation is a multiple of the other.
Key Insight
Most real-world systems are independent: two different conditions that pinpoint one unique outcome, like finding the exact point where supply meets demand in economics.
Definition
For a $2 \times 2$ linear system, independence corresponds to $\det(A) \neq 0$, meaning $A$ is invertible and the unique solution is $x = A^{-1}b$. In general, a consistent system is independent (uniquely solvable) if and only if $\text{rank}(A) = n$ (the number of unknowns). The geometric interpretation is that the hyperplanes defined by the equations intersect at exactly one point.
Example
$A = [[3,1],[1,-2]]$, $\det(A) = -6 - 1 = -7 \neq 0$. Invertible. Solution: $x = A^{-1}b = (1/-7)[[-2,-1],[-1,3]] \cdot b$.
Key Insight
The independence condition $\det(A) \neq 0$ is the exact threshold for Cramer's Rule to apply and for the matrix to have an inverse. Systems that lose independence ($\det = 0$) are on the boundary between the uniquely solvable and the degenerate.