Enter the coefficients for your system of two linear equations:
Enter coefficients and click Solve.
Understanding Linear Systems of Equations
A system of linear equations is a collection of two or more linear equations involving the same set of variables. For a system with two variables (commonly denoted as 'x' and 'y'), it typically takes the form:
The goal of solving such a system is to find the values of 'x' and 'y' that simultaneously satisfy both equations. Geometrically, each linear equation represents a straight line on a 2D Cartesian plane. The solution to the system is the point (x, y) where these two lines intersect.
Methods for Solving Linear Systems
There are several methods to solve a system of linear equations:
Substitution Method: Solve one equation for one variable and substitute that expression into the other equation.
Elimination Method: Multiply one or both equations by constants so that the coefficients of one variable are opposites. Then, add the equations together to eliminate that variable and solve for the remaining one.
Graphical Method: Graph both lines and find the point of intersection.
Matrix Method (Cramer's Rule or Inverse Matrices): This is a more advanced method, often used for larger systems, involving matrix operations.
How This Calculator Works (Using Cramer's Rule)
This calculator uses Cramer's Rule, a method that directly computes the values of the variables using determinants. For a 2×2 system:
a1*x + b1*y = c1 a2*x + b2*y = c2
The determinants are calculated as follows:
Determinant of the coefficient matrix (D): D = (a1 * b2) - (a2 * b1)
Determinant for x (Dx): Replace the x-coefficients (a1, a2) with the constants (c1, c2).
Dx = (c1 * b2) - (c2 * b1)
Determinant for y (Dy): Replace the y-coefficients (b1, b2) with the constants (c1, c2).
Dy = (a1 * c2) - (a2 * c1)
The solutions for x and y are then given by:
If D ≠ 0:
x = Dx / D y = Dy / D
Special Cases:
If D = 0 and Dx = 0 and Dy = 0, the system has infinitely many solutions (the lines are coincident).
If D = 0 and either Dx ≠ 0 or Dy ≠ 0, the system has no solution (the lines are parallel and distinct).
Use Cases
Linear systems of equations are fundamental in many fields:
Engineering: Analyzing electrical circuits, structural loads, and control systems.
Economics: Modeling supply and demand, input-output analysis, and economic forecasting.
Computer Science: Solving problems in graph theory, optimization, and machine learning algorithms.
Physics: Describing motion, forces, and wave phenomena.
Operations Research: Resource allocation and optimization problems.
Statistics: Linear regression analysis.
This calculator provides a quick and accurate way to solve basic 2×2 systems, aiding in understanding, problem-solving, and verification.
function solveSystem() {
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 errorElement = document.getElementById("error");
var resultElement = document.getElementById("result");
errorElement.textContent = ""; // Clear previous errors
// Input validation
if (isNaN(a1) || isNaN(b1) || isNaN(c1) || isNaN(a2) || isNaN(b2) || isNaN(c2)) {
errorElement.textContent = "Please enter valid numbers for all coefficients.";
resultElement.textContent = "";
return;
}
// Calculate determinants
var D = (a1 * b2) – (a2 * b1);
var Dx = (c1 * b2) – (c2 * b1);
var Dy = (a1 * c2) – (a2 * c1);
// Check for special cases
if (D === 0) {
if (Dx === 0 && Dy === 0) {
resultElement.textContent = "Infinitely many solutions (Lines are coincident).";
} else {
resultElement.textContent = "No solution (Lines are parallel and distinct).";
}
} else {
var x = Dx / D;
var y = Dy / D;
resultElement.textContent = "Solution: x = " + x.toFixed(4) + ", y = " + y.toFixed(4);
}
}