Elimination Calculator (2×2 System)
Enter the coefficients of two linear equations to solve for x and y by elimination.
The elimination method solves a system of two linear equations by combining them to cancel one variable. For ax + by = c pairs, the solution is x = (c₁b₂ − c₂b₁) ÷ (a₁b₂ − a₂b₁) and y = (a₁c₂ − a₂c₁) ÷ (a₁b₂ − a₂b₁). If the denominator is zero, there is no single solution.
How to Use This Calculator
Write each equation in the standard form ax + by = c, then enter the three coefficients for each. The first equation uses a₁, b₁ and c₁; the second uses a₂, b₂ and c₂. A term that is missing is just a zero coefficient, and move any constants to the right side first.
The calculator returns the values of x and y that satisfy both equations, and shows the working. If the two equations describe parallel or identical lines, it reports that there is no single solution.
How the Calculation Works
Elimination scales the equations so one variable has matching coefficients, then subtracts to remove it, leaving one equation in one unknown. Carrying that through in general produces the formulas above, which are the same result Cramer's rule gives.
The key quantity is D, the determinant of the coefficients. When D is not zero the lines cross at exactly one point, giving a unique solution. When D is zero the lines are parallel (no solution) or the same line (infinitely many).
D = a₁b₂ − a₂b₁; x = (c₁b₂ − c₂b₁) ÷ D; y = (a₁c₂ − a₂c₁) ÷ D
Worked Example
Solve x + y = 5 and x − y = 1. Here a₁=1, b₁=1, c₁=5, a₂=1, b₂=−1, c₂=1. The determinant is D = (1)(−1) − (1)(1) = −2.
Then x = (5×−1 − 1×1) ÷ −2 = −6 ÷ −2 = 3, and y = (1×1 − 1×5) ÷ −2 = −4 ÷ −2 = 2. So x = 3, y = 2, which satisfies both equations.
Assumptions and Limitations
- Solves two linear equations in two unknowns (x and y)
- A zero determinant means no unique solution — parallel or identical lines
- Equations must be in ax + by = c form; move constants to the right first
- Results are rounded for display; exact answers may be fractions
Who Uses an Elimination Calculator
Algebra students use this to solve a system of two linear equations by elimination. Run it to check homework, confirm a graphing answer, or follow the method step by step when it is not clicking. It is ideal whenever two equations share the same two unknowns and you need the exact intersection point.
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Open calculator →Frequently Asked Questions
What is the elimination method?
It solves a system of equations by adding or subtracting multiples of them to cancel one variable, leaving a single equation in one unknown that you can solve, then back-substituting for the other.
What does it mean when there is no solution?
If the determinant a₁b₂ − a₂b₁ is zero, the two lines are parallel and never meet (no solution) or are the same line (infinitely many solutions). The system has no single unique answer.
Is elimination the same as substitution?
They solve the same systems and give the same answer, but differ in method. Elimination cancels a variable by combining equations; substitution solves one equation for a variable and plugs it into the other.
Can this solve three-variable systems?
No, this tool handles two equations in two unknowns. For three variables, use the RREF calculator, which row-reduces the augmented matrix of a larger system.
How do I write my equations for this tool?
Put each in the form ax + by = c with the variables on the left and the constant on the right. Enter the number in front of x as a, in front of y as b, and the constant as c.
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