Elimination Method Calculator

⚙️ How the Calculator Works

1. Enter the coefficients (a₁, b₁, a₂, b₂) and constants (c₁, c₂) for your two equations.
2. Choose the decimal precision for the solution.
3. The calculator computes the determinant to check for solvability.
4. If solvable, it calculates x and y using the elimination formulas.
5. The result is displayed instantly with the chosen precision.
6. If the system has no solution or infinitely many solutions, a message is displayed instead.

Input validation and real-time updates ensure accurate results for all 2x2 linear systems.

📐 Elimination Method Formulas Explained

The elimination method is a fundamental technique in algebra for solving a system of two linear equations with two variables. This method eliminates one variable by combining the equations, allowing us to solve for the other variable. It is particularly useful when substitution is complex or when equations are given in standard form.

In these formulas:

• a₁, b₁ – coefficients of x and y in the first equation.
• c₁ – constant term in the first equation.
• a₂, b₂ – coefficients of x and y in the second equation.
• c₂ – constant term in the second equation.
• Denominator (a₁·b₂ − a₂·b₁) – the determinant of the system; if this equals zero, the system is either dependent or inconsistent.
• x and y – the solutions of the system of equations.

These formulas are derived from multiplying each equation by appropriate factors to cancel one variable. Once one variable is eliminated, the remaining variable is solved directly, and then substituted back into either original equation to find the second variable.

The elimination formulas can also handle special cases:

• If the determinant is zero and the numerators are zero as well, the system has infinitely many solutions.
• If the determinant is zero but the numerators are non-zero, the system has no solution.
• For all other cases, the system has a unique solution that can be calculated directly using the formulas above.

By including the elimination formulas in the calculator, users can instantly see the solution without manual computation, making this method ideal for students, teachers, and professionals handling algebraic calculations.

📌 Example

For the system:

2x + 3y = 8
4x − y = 2

Step 1: Calculate the determinant:

det = (2·(-1) − 4·3) = -2 − 12 = -14

Step 2: Calculate x using the formula:

x = (8·(-1) − 2·3) / (-14) = (-8 − 6)/(-14) = 14 / 14 = 1

Step 3: Calculate y using the formula:

y = (2·2 − 4·8) / (-14) = (4 − 32)/(-14) = -28 / -14 = 2

Therefore, the solution is x = 1 and y = 2. Using this systematic approach, the elimination method provides a clear, reliable, and quick way to solve any 2x2 system of linear equations.