Solving Systems of Equations Elimination Calculator

Systems of Equations Elimination Calculator

Enter the coefficients and constants for two linear equations in the form:

A₁x + B₁y = C₁

A₂x + B₂y = C₂

Equation 1:

Coefficient of x (A₁): Coefficient of y (B₁): Constant (C₁):

Equation 2:

Coefficient of x (A₂): Coefficient of y (B₂): Constant (C₂):

function calculateElimination() { var a1 = parseFloat(document.getElementById('a1_coeff').value); var b1 = parseFloat(document.getElementById('b1_coeff').value); var c1 = parseFloat(document.getElementById('c1_const').value); var a2 = parseFloat(document.getElementById('a2_coeff').value); var b2 = parseFloat(document.getElementById('b2_coeff').value); var c2 = parseFloat(document.getElementById('c2_const').value); var resultDiv = document.getElementById('result'); resultDiv.innerHTML = "; // Clear previous results // Validate inputs if (isNaN(a1) || isNaN(b1) || isNaN(c1) || isNaN(a2) || isNaN(b2) || isNaN(c2)) { resultDiv.innerHTML = 'Please enter valid numbers for all coefficients and constants.'; return; } // Calculate the determinant (D) of the coefficient matrix var D = (a1 * b2) – (b1 * a2); // Calculate Dx and Dy for Cramer's Rule var Dx = (c1 * b2) – (b1 * c2); var Dy = (a1 * c2) – (c1 * a2); var outputHTML = '

Solution:

'; if (D === 0) { // Determinant is zero, check for no solution or infinitely many solutions if (Dx === 0 && Dy === 0) { outputHTML += 'The system has infinitely many solutions (dependent system).'; outputHTML += 'This means the two equations represent the same line.'; } else { outputHTML += 'The system has no solution (inconsistent system).'; outputHTML += 'This means the two equations represent parallel lines that never intersect.'; } } else { // Unique solution exists var x = Dx / D; var y = Dy / D; outputHTML += 'The system has a unique solution:'; outputHTML += 'x = ' + x.toFixed(4) + ''; outputHTML += 'y = ' + y.toFixed(4) + ''; outputHTML += '(Values rounded to 4 decimal places)'; } resultDiv.innerHTML = outputHTML; } .calculator-container { background-color: #f9f9f9; border: 1px solid #ddd; padding: 20px; border-radius: 8px; max-width: 600px; margin: 20px auto; font-family: Arial, sans-serif; } .calculator-container h2 { text-align: center; color: #333; margin-bottom: 20px; } .calculator-container p { margin-bottom: 10px; line-height: 1.6; } .calculator-container code { background-color: #eef; padding: 2px 4px; border-radius: 4px; font-family: monospace; } .calc-input-group { background-color: #fff; border: 1px solid #eee; padding: 15px; border-radius: 5px; margin-bottom: 15px; } .calc-input-group h3 { margin-top: 0; color: #555; border-bottom: 1px solid #eee; padding-bottom: 10px; margin-bottom: 15px; } .calc-input-group label { display: block; margin-bottom: 5px; font-weight: bold; color: #444; } .calc-input-group input[type="number"] { width: calc(100% – 22px); padding: 10px; margin-bottom: 15px; border: 1px solid #ccc; border-radius: 4px; box-sizing: border-box; } .calculator-container button { display: block; width: 100%; padding: 12px 20px; background-color: #007bff; color: white; border: none; border-radius: 4px; font-size: 18px; cursor: pointer; transition: background-color 0.3s ease; } .calculator-container button:hover { background-color: #0056b3; } .calc-result { margin-top: 20px; padding: 15px; background-color: #e9f7ef; border: 1px solid #d4edda; border-radius: 5px; color: #155724; } .calc-result h3 { color: #155724; margin-top: 0; } .calc-result p { margin-bottom: 5px; } .calc-result .error { color: #721c24; background-color: #f8d7da; border-color: #f5c6cb; padding: 10px; border-radius: 4px; }

Understanding Systems of Linear Equations and the Elimination Method

A system of linear equations consists of two or more linear equations with the same set of variables. When we "solve" such a system, we are looking for the values of the variables that satisfy all equations simultaneously. Geometrically, for a system of two linear equations with two variables (like x and y), finding a solution means finding the point(s) where the lines represented by the equations intersect.

The Elimination Method Explained

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, leaving you with a single equation with one variable that can be easily solved. Once you find the value of one variable, you can substitute it back into one of the original equations to find the value of the other variable.

Steps for the Elimination Method:

Write Equations in Standard Form: Ensure both equations are in the standard form Ax + By = C.
Choose a Variable to Eliminate: Decide whether to eliminate x or y.
Make Coefficients Opposites: Multiply one or both equations by a constant (or constants) so that the coefficients of the variable you chose to eliminate become additive opposites (e.g., +5x and -5x, or +3y and -3y).
Add the Equations: Add the two modified equations together. This step should eliminate one variable, resulting in a single equation with one variable.
Solve for the Remaining Variable: Solve the new equation for the remaining variable.
Substitute Back: Substitute the value you found in step 5 into one of the original equations and solve for the other variable.
Check Your Solution: (Optional but recommended) Substitute both values back into both original equations to ensure they satisfy both.

Example Walkthrough:

Let's solve the following system using the elimination method:

Equation 1: 2x + 3y = 7

Equation 2: 4x - 2y = 2

Standard Form: Both equations are already in standard form.
Choose Variable to Eliminate: Let's choose to eliminate x.
Make Coefficients Opposites:
- The coefficient of x in Eq 1 is 2.
- The coefficient of x in Eq 2 is 4.
- To make them opposites, we can multiply Eq 1 by -2.
New Eq 1: -2 * (2x + 3y) = -2 * 7 => -4x - 6y = -14

Eq 2 remains: 4x - 2y = 2
Add the Equations:
(-4x - 6y) + (4x - 2y) = -14 + 2

-8y = -12
Solve for the Remaining Variable:
y = -12 / -8

y = 3/2 or y = 1.5
Substitute Back: Substitute y = 1.5 into original Eq 1:
2x + 3(1.5) = 7

2x + 4.5 = 7

2x = 7 - 4.5

2x = 2.5

x = 2.5 / 2

x = 1.25
Check Your Solution:
For Eq 1: 2(1.25) + 3(1.5) = 2.5 + 4.5 = 7 (Correct)

For Eq 2: 4(1.25) - 2(1.5) = 5 - 3 = 2 (Correct)

The solution is x = 1.25 and y = 1.5.

Types of Solutions:

Unique Solution: The lines intersect at exactly one point. This is the most common outcome, as shown in the example above.
No Solution (Inconsistent System): The lines are parallel and never intersect. This occurs when, after elimination, you end up with a false statement (e.g., 0 = 5).
Infinitely Many Solutions (Dependent System): The two equations represent the same line. This occurs when, after elimination, you end up with a true statement (e.g., 0 = 0). Any point on the line is a solution.

How to Use the Calculator:

Our Systems of Equations Elimination Calculator simplifies this process. Simply input the coefficients (A₁, B₁, A₂, B₂) and constants (C₁, C₂) for your two linear equations into the respective fields. Click "Solve System," and the calculator will instantly provide the solution (x and y values) or indicate if there are no solutions or infinitely many solutions, based on the principles of the elimination method.