Elimination Calculator – Solve Systems of Equations by EliminationΣelim.calc
Σ ELIMINATION METHOD CALCULATOR
Elimination Calculator
Solve systems of 2 or 3 linear equations by the elimination method with full step-by-step solutions.
// solve_by_elimination()
The Elimination Method Explained
The elimination method (also called the addition method) is one of three standard techniques for solving systems of linear equations, alongside substitution and matrix/Gaussian elimination. It works by multiplying equations by appropriate constants so that when two equations are added together, one variable cancels out — it is “eliminated” — leaving a simpler equation with fewer unknowns.
1
Write both equations in standard form (Ax + By = C). Ensure all variables are on the left side and constants on the right.
2
Choose which variable to eliminate first. Multiply one or both equations by constants so that the chosen variable has equal and opposite coefficients in both equations.
3
Add the equations. The chosen variable cancels out, leaving a single-variable equation that can be solved directly.
4
Substitute back. Use the solved value in one of the original equations to find the remaining variable(s).
5
Check your solution in both original equations to verify correctness.
Σ When to use elimination vs substitution: Elimination works best when coefficients are small integers and one variable can be quickly eliminated. Substitution is easier when one equation already has an isolated variable (e.g., y = 3x + 2). Both methods always produce the same answer.
A system of equations has no solution (is “inconsistent”) when the elimination process produces a contradiction like “0 = 5”. Geometrically, this means the lines are parallel — they have the same slope but different y-intercepts and never intersect. The system of equations is unsolvable in such cases.
A system has infinitely many solutions (is “dependent”) when elimination produces an identity like “0 = 0”. This means both equations represent the same line — they are multiples of each other. Any point on that line is a valid solution. In this case, the solution set is expressed as a general parameterised expression.
Both methods solve the same types of systems and always produce the same answers. Elimination adds/subtracts multiples of equations to cancel a variable. Substitution solves one equation for one variable and substitutes that expression into the other. Elimination is generally faster when coefficients are simple integers; substitution is faster when one variable is already isolated.