Elimination Method
AlgebraThe 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 two equations by adding them together in a way that removes one variable. Once one variable is gone, you solve for the other, then substitute back.
Example
$x + y = 8$ and $x - y = 2$. Add the equations: $2x = 10$, so $x = 5$. Substitute: $5 + y = 8$, so $y = 3$. Solution: $(5, 3)$.
Key Insight
Elimination is like canceling opposites. If one equation has $+y$ and another has $-y$, they disappear when you add, leaving just one variable to solve.
Definition
The elimination method (also called addition method) manipulates equations by multiplying by constants so that one variable's coefficients become opposites, then adds the equations to eliminate that variable. It works best when coefficients are integers and one variable is easy to eliminate.
Example
$3x + 2y = 16$ and $5x - 2y = 0$. Add: $8x = 16$, $x = 2$. Substitute: $3(2) + 2y = 16$, $2y = 10$, $y = 5$. Solution: $(2, 5)$. Check: $5(2) - 2(5) = 0$. Correct.
Key Insight
When coefficients do not cancel directly, multiply one or both equations by constants first. If $2y$ and $3y$ need to cancel, multiply the first equation by $3$ and the second by $2$ to get $6y$ and $6y$ (with opposite signs).
Definition
Elimination is the algorithmic basis of Gaussian elimination for solving $Ax = b$. Each elimination step corresponds to an elementary row operation on the augmented matrix $[A|b]$: replacing row $i$ with row $i + c \cdot (\text{row } j)$. The process reduces $A$ to upper triangular form (row echelon form), after which back-substitution yields the solution. LU decomposition formalizes this process for computational efficiency.
Example
System: $[[2,1],[4,3]][[x],[y]] = [[5],[11]]$. Eliminate $x$: row $2 - 2 \cdot (\text{row } 1)$ gives $[[2,1],[0,1]][[x],[y]] = [[5],[1]]$. Back-substitute: $y = 1$, $x = (5-1)/2 = 2$.
Key Insight
Gaussian elimination runs in $O(n^3)$ time for $n$ equations. For sparse systems (common in physics simulations), specialized algorithms like conjugate gradient exploit the structure to run much faster.