Input the coefficients and constants for two linear equations of the form:
a1*x + b1*y = c1
a2*x + b2*y = c2
Understanding Systems of Linear Equations
A system of linear equations involves two or more linear equations with the same set of unknown variables. In this calculator, we focus on systems with two variables, typically denoted as 'x' and 'y'. A general form for such a system is:
a₁x + b₁y = c₁
a₂x + b₂y = c₂
Here, 'a₁', 'b₁', 'c₁', 'a₂', 'b₂', and 'c₂' are known coefficients and constants, while 'x' and 'y' are the unknown variables we aim to find. The solution to such a system represents the point (or points) where the lines represented by these equations intersect on a graph.
Methods for Solving
There are several algebraic methods to solve systems of linear equations. This calculator employs the Cramer's Rule method, which is efficient for systems of two variables and relies on determinants.
Cramer's Rule Explained
Cramer's Rule involves calculating three determinants:
D (Determinant of the coefficient matrix):
D = a₁b₂ – a₂b₁
Dx (Determinant with x-coefficients replaced by constants):
Dx = c₁b₂ – c₂b₁
Dy (Determinant with y-coefficients replaced by constants):
Dy = a₁c₂ – a₂c₁
The solutions for x and y are then found by:
If D ≠ 0:
x = Dx / D
y = Dy / D
If D = 0:
If Dx = 0 and Dy = 0, the system has infinitely many solutions (the lines are coincident).
If Dx ≠ 0 or Dy ≠ 0, the system has no solution (the lines are parallel and distinct).
Interpreting the Results
Unique Solution: When the determinant D is not zero, there is exactly one pair of (x, y) values that satisfies both equations. This corresponds to the single intersection point of two distinct, non-parallel lines.
No Solution: If D is zero, but either Dx or Dy is non-zero, it means the lines are parallel and never intersect. There is no (x, y) pair that satisfies both equations simultaneously.
Infinitely Many Solutions: If D, Dx, and Dy are all zero, the two equations represent the exact same line. Every point on this line is a solution to the system.
Use Cases
Systems of linear equations are fundamental in various fields:
Engineering: Analyzing circuits, structural mechanics, and fluid dynamics.
Economics: Modeling supply and demand, market equilibrium, and resource allocation.
Computer Graphics: Transformations, projections, and solving for points and vectors.
Physics: Solving problems involving forces, motion, and energy.
Operations Research: Optimization problems, linear programming.
Everyday Problem Solving: Calculating costs based on different quantities, or determining mixtures.
This calculator provides a quick and accurate way to find the solutions for two-variable linear systems, aiding in understanding and solving problems across these diverse domains.
function solveEquations() {
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 resultDiv = document.getElementById("result");
var errorDiv = document.getElementById("errorMessage");
resultDiv.innerHTML = ""; // Clear previous results
errorDiv.innerHTML = ""; // Clear previous errors
// Check if all inputs are valid numbers
if (isNaN(a1) || isNaN(b1) || isNaN(c1) || isNaN(a2) || isNaN(b2) || isNaN(c2)) {
errorDiv.innerHTML = "Please enter valid numbers for all coefficients and constants.";
return;
}
// Calculate determinants using Cramer's Rule
var D = a1 * b2 – a2 * b1;
var Dx = c1 * b2 – c2 * b1;
var Dy = a1 * c2 – a2 * c1;
var output = "";
if (D !== 0) {
// Unique solution
var x = Dx / D;
var y = Dy / D;
// Format to a reasonable number of decimal places if not integers
if (Number.isInteger(x) && Number.isInteger(y)) {
output = "Unique Solution:x = " + x + "y = " + y;
} else {
output = "Unique Solution:x = " + x.toFixed(4) + "y = " + y.toFixed(4);
}
} else {
// D = 0, check for no solution or infinite solutions
if (Dx === 0 && Dy === 0) {
output = "Infinitely Many Solutions (the equations represent the same line).";
} else {
output = "No Solution (the lines are parallel and distinct).";
}
}
resultDiv.innerHTML = output;
}