Solve systems of linear equations with up to three variables.
Equation 1
x +
y +
z =
Equation 2
x +
y +
z =
Equation 3 (Optional)
x +
y +
z =
Understanding Linear Systems
A system of linear equations is a collection of two or more linear equations involving the same set of variables. These variables are typically denoted by letters like x, y, and z. Each equation in the system represents a line (in 2D), a plane (in 3D), or a hyperplane (in higher dimensions).
What is a Linear Equation?
A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. For example, 2x + 3y = 7 is a linear equation with variables x and y. The general form of a linear equation in n variables is:
a₁x₁ + a₂x₂ + ... + anxn = b
where a₁, a₂, ..., an are the coefficients (constants), x₁, x₂, ..., xn are the variables, and b is the constant term.
Solving Linear Systems
The goal when solving a system of linear equations is to find a set of values for the variables that satisfies all equations in the system simultaneously. Geometrically, this corresponds to finding the point(s) of intersection of the lines, planes, or hyperplanes represented by the equations.
Methods for Solving:
Substitution Method: Solve one equation for one variable and substitute that expression into the other equations.
Elimination Method (or Addition Method): Manipulate the equations (by multiplying them by constants) so that adding or subtracting them eliminates one or more variables.
Matrix Methods (e.g., Gaussian Elimination, Cramer's Rule): Represent the system using matrices and apply matrix operations to find the solution. This is particularly useful for larger systems.
This Calculator's Approach (for 2×2 and 3×3 Systems)
This calculator implements a common algebraic method, often a variation of Gaussian elimination or Cramer's rule using determinants for 2×2 and 3×3 systems. It takes the coefficients of the variables (x, y, z) and the constant terms for each equation and calculates the unique solution, if one exists.
The solution can also be found using determinants (Cramer's Rule), though calculating the 3×3 determinants can be complex. The calculator handles these calculations internally.
Possible Outcomes:
Unique Solution: The system intersects at a single point (e.g., x=1, y=2, z=3). This occurs when the determinant of the coefficient matrix is non-zero.
No Solution (Inconsistent System): The equations represent parallel lines or planes that never intersect. This occurs when the determinant is zero, but the determinants for the variables (Dx, Dy, etc.) are non-zero.
Infinitely Many Solutions (Dependent System): The equations represent the same line or plane, or lines/planes that intersect along a line or plane. This occurs when all determinants (D, Dx, Dy, etc.) are zero.
Use Cases:
Engineering & Physics: Analyzing circuits, mechanics, and fluid dynamics.
Economics: Modeling supply and demand, input-output analysis.
Computer Graphics: Transformations and projections.
Operations Research: Optimization problems.
Mathematics Education: Understanding and visualizing algebraic concepts.
function solveLinearSystem() {
// — Input Retrieval —
var eq1_a1 = parseFloat(document.getElementById('eq1_a1').value);
var eq1_a2 = parseFloat(document.getElementById('eq1_a2').value);
var eq1_a3 = parseFloat(document.getElementById('eq1_a3').value);
var eq1_b = parseFloat(document.getElementById('eq1_b').value);
var eq2_a1 = parseFloat(document.getElementById('eq2_a1').value);
var eq2_a2 = parseFloat(document.getElementById('eq2_a2').value);
var eq2_a3 = parseFloat(document.getElementById('eq2_a3').value);
var eq2_b = parseFloat(document.getElementById('eq2_b').value);
var eq3_a1 = parseFloat(document.getElementById('eq3_a1').value);
var eq3_a2 = parseFloat(document.getElementById('eq3_a2').value);
var eq3_a3 = parseFloat(document.getElementById('eq3_a3').value);
var eq3_b = parseFloat(document.getElementById('eq3_b').value);
var resultDiv = document.getElementById('result');
resultDiv.innerHTML = "; // Clear previous results
// — Input Validation —
var inputs = [eq1_a1, eq1_a2, eq1_a3, eq1_b,
eq2_a1, eq2_a2, eq2_a3, eq2_b,
eq3_a1, eq3_a2, eq3_a3, eq3_b];
var isValid = true;
for (var i = 0; i < inputs.length; i++) {
if (isNaN(inputs[i])) {
isValid = false;
break;
}
}
if (!isValid) {
resultDiv.innerHTML = 'Please enter valid numbers for all coefficients and constants.';
resultDiv.style.backgroundColor = '#dc3545'; // Error color
return;
}
// — Determine System Size —
// Check if the third equation is effectively zero (all coefficients and constant are 0)
var is3x3 = !(eq3_a1 === 0 && eq3_a2 === 0 && eq3_a3 === 0 && eq3_b === 0);
if (is3x3) {
solve3x3([eq1_a1, eq1_a2, eq1_a3, eq1_b],
[eq2_a1, eq2_a2, eq2_a3, eq2_b],
[eq3_a1, eq3_a2, eq3_a3, eq3_b]);
} else {
solve2x2([eq1_a1, eq1_a2, eq1_b],
[eq2_a1, eq2_a2, eq2_b]);
}
}
function solve2x2(eq1, eq2) {
var a11 = eq1[0], a12 = eq1[1], b1 = eq1[2];
var a21 = eq2[0], a22 = eq2[1], b2 = eq2[2];
var determinant = a11 * a22 – a12 * a21;
var resultDiv = document.getElementById('result');
if (determinant === 0) {
// Check for no solution vs. infinite solutions
// If a11*b2 == a21*b1 AND a12*b2 == a22*b1, then infinite solutions
// Otherwise, no solution
if (Math.abs(a11 * b2 – a21 * b1) < 1e-9 && Math.abs(a12 * b2 – a22 * b1) 1e-9 || Math.abs(detDy) > 1e-9 || Math.abs(detDz) > 1e-9) {
resultDiv.innerHTML = 'No Solution (Inconsistent System)';
resultDiv.style.backgroundColor = '#dc3545'; // Error color
} else {
// D = 0 and Dx = Dy = Dz = 0. This indicates infinite solutions.
resultDiv.innerHTML = 'Infinite Solutions (Dependent System)';
resultDiv.style.backgroundColor = '#ffc107'; // Warning color
}
} else {
// Calculate determinants for variables (Dx, Dy, Dz)
var detDx = b1 * (a22 * a33 – a23 * a32) –
a12 * (b2 * a33 – a23 * b3) +
a13 * (b2 * a32 – a22 * b3);
var detDy = a11 * (b2 * a33 – a23 * b3) –
b1 * (a21 * a33 – a23 * a31) +
a13 * (a21 * b3 – b2 * a31);
var detDz = a11 * (a22 * b3 – b2 * a32) –
a12 * (a21 * b3 – b2 * a31) +
b1 * (a21 * a32 – a22 * a31);
// Calculate the variables
var x = detDx / determinant;
var y = detDy / determinant;
var z = detDz / determinant;
resultDiv.innerHTML = 'Solution: x = ' + x.toFixed(4) + ', y = ' + y.toFixed(4) + ', z = ' + z.toFixed(4) + ";
resultDiv.style.backgroundColor = '#28a745'; // Success color
}
}