Enter the coefficients and constants for your system of three linear equations with three variables (x, y, z) below. The calculator will solve for x, y, and z using Gauss-Jordan elimination.
The system is represented as:
a11x + a12y + a13z = b1
a21x + a22y + a23z = b2
a31x + a32y + a33z = b3
.calculator-container {
font-family: 'Segoe UI', Tahoma, Geneva, Verdana, sans-serif;
background-color: #f9f9f9;
border: 1px solid #ddd;
border-radius: 8px;
padding: 20px;
max-width: 600px;
margin: 20px auto;
box-shadow: 0 4px 8px rgba(0, 0, 0, 0.05);
}
.calculator-container h2 {
color: #333;
text-align: center;
margin-bottom: 20px;
}
.calculator-container p {
color: #555;
line-height: 1.6;
margin-bottom: 10px;
}
.input-group {
display: flex;
flex-wrap: wrap;
align-items: center;
margin-bottom: 15px;
gap: 10px;
}
.input-group label {
flex: 0 0 auto;
width: 40px;
color: #333;
font-weight: bold;
text-align: right;
}
.input-group input[type="number"] {
flex: 1 1 80px;
padding: 10px;
border: 1px solid #ccc;
border-radius: 4px;
font-size: 1em;
max-width: 100px;
}
.calculator-container button {
display: block;
width: 100%;
padding: 12px 20px;
background-color: #007bff;
color: white;
border: none;
border-radius: 4px;
font-size: 1.1em;
cursor: pointer;
transition: background-color 0.3s ease;
margin-top: 20px;
}
.calculator-container button:hover {
background-color: #0056b3;
}
.calculator-result {
margin-top: 25px;
padding: 15px;
background-color: #e9ecef;
border: 1px solid #dee2e6;
border-radius: 4px;
color: #333;
font-size: 1.1em;
text-align: center;
min-height: 50px;
display: flex;
flex-direction: column;
justify-content: center;
align-items: center;
}
.calculator-result h3 {
color: #007bff;
margin-top: 0;
margin-bottom: 10px;
}
.calculator-result p {
margin: 5px 0;
font-weight: bold;
}
function calculateAugmentedMatrix() {
// Get input values
var a11 = parseFloat(document.getElementById('a11').value);
var a12 = parseFloat(document.getElementById('a12').value);
var a13 = parseFloat(document.getElementById('a13').value);
var b1 = parseFloat(document.getElementById('b1').value);
var a21 = parseFloat(document.getElementById('a21').value);
var a22 = parseFloat(document.getElementById('a22').value);
var a23 = parseFloat(document.getElementById('a23').value);
var b2 = parseFloat(document.getElementById('b2').value);
var a31 = parseFloat(document.getElementById('a31').value);
var a32 = parseFloat(document.getElementById('a32').value);
var a33 = parseFloat(document.getElementById('a33').value);
var b3 = parseFloat(document.getElementById('b3').value);
// Validate inputs
if (isNaN(a11) || isNaN(a12) || isNaN(a13) || isNaN(b1) ||
isNaN(a21) || isNaN(a22) || isNaN(a23) || isNaN(b2) ||
isNaN(a31) || isNaN(a32) || isNaN(a33) || isNaN(b3)) {
document.getElementById('result').innerHTML = 'Please enter valid numbers for all matrix elements.';
return;
}
// Create the augmented matrix
var matrix = [
[a11, a12, a13, b1],
[a21, a22, a23, b2],
[a31, a32, a33, b3]
];
var n = 3; // Number of variables/rows
// Gauss-Jordan Elimination
for (var i = 0; i < n; i++) {
// Find pivot row (for numerical stability)
var maxRow = i;
for (var k = i + 1; k Math.abs(matrix[maxRow][i])) {
maxRow = k;
}
}
// Swap rows if necessary
var temp = matrix[i];
matrix[i] = matrix[maxRow];
matrix[maxRow] = temp;
// Check for singular matrix (no unique solution or infinite solutions)
if (matrix[i][i] === 0) {
// If the pivot element is zero after trying to swap, the matrix is singular.
// This means either no solution or infinite solutions.
// Check if any row from current pivot onwards is inconsistent (e.g., [0 0 0 | non-zero])
var isConsistent = true;
for (var rowCheck = i; rowCheck < n; rowCheck++) {
var allCoeffsZero = true;
for (var colCheck = 0; colCheck < n; colCheck++) {
if (matrix[rowCheck][colCheck] !== 0) {
allCoeffsZero = false;
break;
}
}
if (allCoeffsZero && matrix[rowCheck][n] !== 0) {
isConsistent = false; // Found a row like [0 0 0 | non-zero]
break;
}
}
if (!isConsistent) {
document.getElementById('result').innerHTML = 'No solution exists for this system.';
} else {
document.getElementById('result').innerHTML = 'Infinite solutions or system is dependent.';
}
return;
}
// Normalize the pivot row (make pivot element 1)
var pivot = matrix[i][i];
for (var j = i; j <= n; j++) {
matrix[i][j] /= pivot;
}
// Eliminate other rows (make elements in current column zero)
for (var k = 0; k < n; k++) {
if (k !== i) {
var factor = matrix[k][i];
for (var j = i; j <= n; j++) {
matrix[k][j] -= factor * matrix[i][j];
}
}
}
}
// Extract solutions
var x = matrix[0][n];
var y = matrix[1][n];
var z = matrix[2][n];
// Display results
var resultHTML = '
An augmented matrix is a compact way to represent a system of linear equations. It's formed by combining the coefficient matrix of a system of equations with its constant vector. For a system like:
a11x + a12y + a13z = b1
a21x + a22y + a23z = b2
a31x + a32y + a33z = b3
The augmented matrix would be written as:
[ a11 a12 a13 | b1 ]
[ a21 a22 a23 | b2 ]
[ a31 a32 a33 | b3 ]
The vertical line separates the coefficient matrix from the constant terms.
Purpose of Augmented Matrices
The primary purpose of an augmented matrix is to simplify the process of solving systems of linear equations. By representing the system in this matrix form, we can apply systematic operations, known as elementary row operations, to transform the matrix into a simpler form (like row echelon form or reduced row echelon form) from which the solution can be easily read.
How the Calculator Works (Gauss-Jordan Elimination)
This calculator uses the Gauss-Jordan elimination method. This method involves performing a series of elementary row operations to transform the augmented matrix into its reduced row echelon form. In this form, the coefficient part of the matrix becomes an identity matrix (1s on the main diagonal, 0s elsewhere), and the constant part directly gives the solution for each variable.
The elementary row operations are:
Swapping two rows.
Multiplying a row by a non-zero scalar.
Adding a multiple of one row to another row.
The calculator systematically applies these operations to achieve the reduced row echelon form, thereby solving for x, y, and z. It also identifies cases where there is no unique solution (inconsistent system) or infinitely many solutions (dependent system).
Example Usage:
Consider the system of equations:
x + y + z = 6
2x + y + z = 7
x + 2y + z = 8
Input the coefficients and constants into the calculator:
a11=1, a12=1, a13=1, b1=6
a21=2, a22=1, a23=1, b2=7
a31=1, a32=2, a33=1, b3=8
Click "Solve System", and the calculator will output the solution: x = 1, y = 2, z = 3.