Solve a system of two linear equations with two variables (x and y).
Solution:
Enter coefficients and constants above.
Understanding Linear Systems and Their Solutions
A system of linear equations is a set of two or more linear equations containing the same set of variables. In this calculator, we focus on a system of two linear equations with two variables, commonly denoted as 'x' and 'y'. These equations represent lines on a 2D Cartesian plane.
A general system of two linear equations can be written as:
Here, a₁, b₁, c₁, a₂, b₂, and c₂ are known constants, and x and y are the variables we aim to solve for.
Methods for Solving Linear Systems
There are several methods to solve such systems, including:
Substitution Method: Solve one equation for one variable and substitute that expression into the other equation.
Elimination Method (or Addition 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.
Graphical Method: Graph both lines and find the point of intersection. This method is often less precise for non-integer solutions.
Matrix Method (Cramer's Rule or Inverse Matrices): Represent the system in matrix form and use matrix operations. This is particularly useful for larger systems.
How This Calculator Works (Cramer's Rule)
This calculator uses Cramer's Rule, a method that utilizes determinants to find the solution to a system of linear equations. For a 2×2 system, the solution is given by:
First, we calculate the determinant of the coefficient matrix (D):
D = a₁b₂ - a₂b₁
If D is not zero, the system has a unique solution. We then calculate:
Dₓ = c₁b₂ - c₂b₁ (Replace x coefficients with constants)
Dy = a₁c₂ - a₂c₁ (Replace y coefficients with constants)
The solution is then:
x = Dₓ / D
y = Dy / D
Possible Outcomes:
Unique Solution: If D ≠ 0, there is exactly one pair (x, y) that satisfies both equations.
No Solution: If D = 0 AND (Dₓ ≠ 0 OR Dy ≠ 0), the lines are parallel and distinct, meaning there is no point of intersection.
Infinitely Many Solutions: If D = 0 AND Dₓ = 0 AND Dy = 0, the two equations represent the same line, and any point on the line is a solution.
Use Cases for Linear Systems
Systems of linear equations are fundamental in many fields:
This calculator provides a quick and accurate way to find the solutions for basic 2×2 linear systems, helping to understand the relationships between variables in various mathematical and scientific contexts.
function solveLinearSystem() {
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 errorDiv = document.getElementById("error");
var solutionDiv = document.getElementById("solution");
errorDiv.textContent = ""; // Clear previous errors
solutionDiv.textContent = ""; // Clear previous solutions
// Validate inputs
if (isNaN(a1) || isNaN(b1) || isNaN(c1) || isNaN(a2) || isNaN(b2) || isNaN(c2)) {
errorDiv.textContent = "Error: Please enter valid numbers for all coefficients and constants.";
return;
}
// Calculate the determinant of the coefficient matrix
var determinant = (a1 * b2) – (a2 * b1);
if (determinant === 0) {
// Check for no solution or infinite solutions
var determinantX = (c1 * b2) – (c2 * b1);
var determinantY = (a1 * c2) – (a2 * c1);
if (determinantX === 0 && determinantY === 0) {
solutionDiv.textContent = "Infinite Solutions (The equations represent the same line).";
solutionDiv.style.color = "#007bff"; // Blue for informational message
} else {
solutionDiv.textContent = "No Solution (The lines are parallel and distinct).";
solutionDiv.style.color = "#dc3545"; // Red for error/no solution
}
} else {
// Calculate determinants for x and y
var determinantX = (c1 * b2) – (c2 * b1);
var determinantY = (a1 * c2) – (a2 * c1);
// Calculate the solution
var x = determinantX / determinant;
var y = determinantY / determinant;
solutionDiv.textContent = "x = " + x.toFixed(4) + ", y = " + y.toFixed(4);
solutionDiv.style.color = "#28a745"; // Green for success
}
}