Elimination Method Calculator

Elimination Method Calculator for Systems of Linear Equations

Use this calculator to solve a system of two linear equations with two variables (x and y) using the elimination method. Enter the coefficients and constants for each equation in the standard form:

Equation 1: A1x + B1y = C1

Equation 2: A2x + B2y = C2

Equation 1:

Equation 2:

Understanding the Elimination Method

The elimination method, also known as the addition method, is a powerful algebraic technique used to solve systems of linear equations. The core idea is to eliminate one of the variables by adding or subtracting the equations, thereby reducing the system to a single equation with one variable, which is then easy to solve.

How the Elimination Method Works:

1. Standard Form: Ensure both equations are in the standard form Ax + By = C.
2. Choose a Variable to Eliminate: Decide whether to eliminate 'x' or 'y'.
3. Multiply Equations (if necessary): Multiply one or both equations by a non-zero number so that the coefficients of the variable you chose to eliminate become opposites (e.g., 3x and -3x) or identical (e.g., 3x and 3x).
4. Add or Subtract Equations:
   • If the coefficients are opposites, add the two equations together.
   • If the coefficients are identical, subtract one equation from the other.
   This step should eliminate one variable, leaving you with a single equation in one variable.
5. Solve for the Remaining Variable: Solve the resulting equation for the remaining variable.
6. Substitute Back: Substitute the value you found in step 5 back into one of the original equations to solve for the other variable.
7. Check Your Solution: Substitute both values back into both original equations to ensure they satisfy both equations.

Example:

Let's solve the system:

Equation 1: 2x + 3y = 12

Equation 2: 5x - 2y = 11

1. Choose to eliminate 'y'. The coefficients are 3 and -2. The least common multiple is 6.
2. Multiply Equation 1 by 2: 2 * (2x + 3y) = 2 * 12 → 4x + 6y = 24
3. Multiply Equation 2 by 3: 3 * (5x - 2y) = 3 * 11 → 15x - 6y = 33
4. Add the new equations:

     4x + 6y = 24
   + 15x - 6y = 33
   ----------------
     19x      = 57

5. Solve for x: 19x = 57 → x = 57 / 19 → x = 3
6. Substitute x = 3 into Equation 1:

   2(3) + 3y = 12
   6 + 3y = 12
   3y = 12 - 6
   3y = 6
   y = 6 / 3
   y = 2

7. Solution: x = 3, y = 2.

Special Cases:

• No Solution: If, during the elimination process, you end up with a false statement (e.g., 0 = 5), it means the lines are parallel and never intersect. There is no solution to the system.
• Infinitely Many Solutions: If you end up with a true statement (e.g., 0 = 0), it means the two equations represent the same line. There are infinitely many solutions, as every point on the line is a solution.

This calculator uses a method derived from the elimination process (Cramer's Rule) to efficiently handle these cases and provide accurate solutions.

Infinitely Many Solutions:

No Solution:

Solution: