Solve a system of two linear equations with two variables (x and y).
Equation 1:
Equation 2:
Solution:
Enter values to see the solution
Understanding Simultaneous Equations and How This Calculator Works
A system of simultaneous linear equations involves two or more equations with the same set of unknown variables. When we talk about solving for two variables, say 'x' and 'y', we typically have two linear equations. The goal is to find the unique values of 'x' and 'y' that satisfy *both* equations simultaneously. Geometrically, this represents finding the point of intersection of two lines on a 2D plane.
The General Form
The two equations are usually represented in the general form:
Equation 1: a1*x + b1*y = c1
Equation 2: a2*x + b2*y = c2
Where a1, b1, c1, a2, b2, and c2 are known coefficients and constants.
How the Calculator Solves Them (Using Cramer's Rule)
This calculator uses a common and robust method for solving systems of linear equations called Cramer's Rule. It involves calculating determinants of matrices.
1. The Determinant of the Coefficient Matrix (D):
This is calculated from the coefficients of 'x' and 'y':
D = (a1 * b2) - (a2 * b1)
2. Determinant for x (Dx):
Replace the 'x' coefficients (a1, a2) with the constants (c1, c2):
Dx = (c1 * b2) - (c2 * b1)
3. Determinant for y (Dy):
Replace the 'y' coefficients (b1, b2) with the constants (c1, c2):
Dy = (a1 * c2) - (a2 * c1)
4. Finding x and y:
If the determinant D is not zero, a unique solution exists:
x = Dx / D
y = Dy / D
Special Cases:
If D = 0 and Dx = 0 and Dy = 0: The system has infinitely many solutions (the two equations represent the same line).
If D = 0 and (Dx != 0 or Dy != 0): The system has no solution (the two lines are parallel and never intersect).
Use Cases:
Simultaneous equations are fundamental in many fields:
Physics: Analyzing circuits, mechanics, and motion.
Economics: Modeling supply and demand, market equilibrium.
Computer Graphics: Transformations and projections.
Everyday Problems: Calculating costs of multiple items when total quantities and costs are known, or solving rate problems.
function calculateSimultaneousEquations() {
var a1 = parseFloat(document.getElementById("a1").value);
var b1 = parseFloat(document.getElementById("b1").value);
var c1 = parseFloat(document.getElementById("c1").value);
var a2 = parseFloat(document.getElementById("a2").value);
var b2 = parseFloat(document.getElementById("b2").value);
var c2 = parseFloat(document.getElementById("c2").value);
var solutionTextElement = document.getElementById("solutionText");
// Check if inputs are valid numbers
if (isNaN(a1) || isNaN(b1) || isNaN(c1) || isNaN(a2) || isNaN(b2) || isNaN(c2)) {
solutionTextElement.textContent = "Please enter valid numbers for all coefficients and constants.";
solutionTextElement.style.color = "#dc3545"; // Red for error
return;
}
// Calculate the main determinant (D)
var determinant = (a1 * b2) – (a2 * b1);
// Calculate determinants for x (Dx) and y (Dy)
var detX = (c1 * b2) – (c2 * b1);
var detY = (a1 * c2) – (a2 * c1);
// Determine the solution based on the determinants
if (determinant === 0) {
if (detX === 0 && detY === 0) {
solutionTextElement.textContent = "Infinitely many solutions (lines are coincident).";
solutionTextElement.style.color = "#ffc107"; // Yellow for warning/special case
} else {
solutionTextElement.textContent = "No solution (lines are parallel).";
solutionTextElement.style.color = "#dc3545"; // Red for error
}
} else {
var x = detX / determinant;
var y = detY / determinant;
solutionTextElement.textContent = "x = " + x.toFixed(4) + ", y = " + y.toFixed(4);
solutionTextElement.style.color = "#28a745"; // Green for success
}
}