Elimination Method Calculator — Systems of Equations
Solve any system of two linear equations (a₁x + b₁y = c₁ and a₂x + b₂y = c₂) using the elimination method — enter the six coefficients and get x, y, a step-by-step solution, and a plot of both lines.
Write the system of equations
Multiply Eq1 by the leading coefficient of Eq2, and vice versa
Subtract to eliminate x
Substitute y back into Eq1 to find x
- 1
Determinant D = a₁b₂ − a₂b₁
2 × -1 − 4 × 3 = -14 - 2
x numerator = c₁b₂ − c₂b₁
12 × -1 − 5 × 3 = -27 - 3
x = (c₁b₂ − c₂b₁) ÷ D
-27 ÷ -14 = 1.9286 - 4
y numerator = a₁c₂ − a₂c₁
2 × 5 − 4 × 12 = -38 - 5
y = (a₁c₂ − a₂c₁) ÷ D
-38 ÷ -14 = 2.7143Both x and y simultaneously satisfy both original equations.
How does this calculator work?
Solve a₁x + b₁y = c₁ and a₂x + b₂y = c₂ by elimination: multiply each equation so one variable cancels, solve the remaining single-variable equation, then substitute back. The determinant D = a₁b₂ − a₂b₁ must be non-zero for a unique solution.
Formula
How this is calculated
The elimination method removes one variable by multiplying each equation by a chosen factor and then adding or subtracting the results. For example, to eliminate x from a₁x + b₁y = c₁ and a₂x + b₂y = c₂, multiply the first equation by a₂ and the second by a₁, then subtract. The x terms cancel, leaving a single equation in y alone. Solving for y and substituting back gives x.
The same result follows from Cramer's rule: compute the determinant D = a₁b₂ − a₂b₁, then x = (c₁b₂ − c₂b₁)/D and y = (a₁c₂ − a₂c₁)/D. If D = 0 the lines are parallel (no solution) or coincident (infinite solutions).
Geometrically, each linear equation defines a straight line on the xy-plane. The solution is the point where they cross — so if the lines are parallel they never cross (no solution), and if they are the same line every point satisfies both equations (infinite solutions). The plot shows both lines and marks the intersection.
Frequently asked questions
Elimination is usually faster when the coefficients of one variable have a simple relationship — for example, when one coefficient is a multiple of the other, the multiplication step is trivial. Substitution is often easier when one equation already has a variable isolated (e.g. y = 2x + 3). Both methods always give the same answer.
A determinant D = 0 means the two lines have the same slope (they are parallel or identical). If they are parallel the system has no solution. If they are the same line every point on it is a solution — infinitely many solutions. In either case the equations cannot be solved for a unique (x, y).
No — it solves only 2 × 2 systems (two equations, two unknowns). For three equations and three unknowns you would need Gaussian elimination or a 3 × 3 determinant, which is a separate operation beyond the scope of this tool.
Also known as
TG we-Calculate Editorial Team. (2026). Elimination Method Calculator — Systems of Equations [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/elimination-method-calculator
TG we-Calculate Editorial Team. "Elimination Method Calculator — Systems of Equations." TG we-Calculate. 2026. https://we-calculate.com/calculator/elimination-method-calculator.
TG we-Calculate Editorial Team, "Elimination Method Calculator — Systems of Equations," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/elimination-method-calculator
@misc{wecalculate_elimination_method_calculator, title = {Elimination Method Calculator — Systems of Equations}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/elimination-method-calculator}}, year = {2026}, note = {TG we-Calculate} }
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