Elimination Method
The elimination method solves a system of equations by adding or subtracting equations to cancel out one variable, then solving for the remaining variable.
Definition
The elimination method solves a system by adding or subtracting the equations, after multiplying by constants if needed, so that one variable's coefficients become opposites and cancel out, leaving a single equation in one variable to solve, then substituting back for the other. It works best when coefficients are integers and one variable is easy to eliminate; when coefficients do not cancel directly, multiply one or both equations first so the target coefficients match with opposite signs. Elimination is the algorithmic basis of Gaussian elimination for solving $Ax = b$: each step corresponds to an elementary row operation on the augmented matrix $[A|b]$, replacing row $i$ with row $i + c \cdot (\text{row } j)$, reducing $A$ to row echelon form so that back-substitution yields the solution; LU decomposition formalizes this process for computational efficiency.
Example
For $x + y = 8$ and $x - y = 2$, adding gives $2x = 10$, so $x = 5$ and $y = 3$. For $3x + 2y = 16$ and $5x - 2y = 0$, adding gives $8x = 16$, so $x = 2$ and $y = 5$. In matrix form, the system $[[2,1],[4,3]][[x],[y]] = [[5],[11]]$ eliminates $x$ via row $2 - 2 \cdot (\text{row } 1)$, giving $[[2,1],[0,1]][[x],[y]] = [[5],[1]]$, so $y = 1$ and $x = 2$ by back-substitution.
Key Insight
Elimination is like canceling opposites: if one equation has $+y$ and another has $-y$, they disappear when added, leaving one variable to solve. Gaussian elimination, the general form of this idea, runs in $O(n^3)$ time for $n$ equations; for the large sparse systems common in physics simulations, specialized algorithms like conjugate gradient exploit the structure to run much faster.