This calculator solves a system of two linear equations with two variables (x and y) using the substitution or elimination method. Enter the coefficients and constants for each equation.
Understanding Systems of Linear Equations
A system of linear equations is a collection of two or more linear equations involving the same set of variables. In this calculator, we focus on a system of two linear equations with two variables, typically denoted as 'x' and 'y'. Such a system can be represented in the general form:
Where a1, b1, c1, a2, b2, and c2 are constants (coefficients and terms).
Methods for Solving
There are several algebraic methods to solve systems of linear equations, including:
Substitution Method: Solve one equation for one variable, then substitute that expression into the other equation. This reduces the system to a single equation with one variable.
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, resulting in a single equation with one variable.
Graphical Method: Graph both lines represented by the equations. The point where the lines intersect is the solution to the system.
This calculator uses an algebraic approach (conceptually similar to elimination or Cramer's rule) derived from the general forms to find the exact values of 'x' and 'y' that satisfy both equations simultaneously.
Mathematical Derivation (Determinant Method)
A common way to solve a 2×2 system algebraically involves determinants. The system is:
a1*x + b1*y = c1 a2*x + b2*y = c2
The determinant of the coefficient matrix (D) is:
D = a1*b2 - a2*b1
If D is not zero, there is a unique solution. The determinants for x (Dx) and y (Dy) are:
Dx = c1*b2 - c2*b1Dy = a1*c2 - a2*c1
The solution is then:
x = Dx / Dy = Dy / D
Interpreting the Results
Unique Solution: If the determinant D is non-zero, there is exactly one pair of (x, y) values that satisfies both equations. This means the two lines represented by the equations intersect at a single point.
No Solution: If D = 0 and either Dx or Dy (or both) is non-zero, the system has no solution. This occurs when the two lines are parallel and distinct; they never intersect.
Infinitely Many Solutions: If D = 0 and both Dx = 0 and Dy = 0, the system has infinitely many solutions. This happens when the two equations represent the same line; every point on the line is a solution.
Use Cases
Systems of linear equations are fundamental in many fields:
Economics: Modeling supply and demand, calculating equilibrium prices.
Computer Graphics: Transformations, projections.
Physics: Kinematics, dynamics problems.
Everyday Budgeting: Balancing multiple accounts or financial goals with constraints.
This calculator provides a quick and accurate way to find these solutions, aiding in problem-solving and understanding the relationships between variables.
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 resultDiv = document.getElementById("result");
resultDiv.innerHTML = ""; // Clear previous results
var inputs = [a1, b1, c1, a2, b2, c2];
for (var i = 0; i < inputs.length; i++) {
if (isNaN(inputs[i])) {
resultDiv.innerHTML = "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
// If a1*c2 – a2*c1 = 0 and c1*b2 – c2*b1 = 0 (and determinant is 0),
// it implies the lines are coincident (infinite solutions).
// Otherwise, if determinant is 0 but these aren't, lines are parallel (no solution).
if ((a1 * c2 – a2 * c1 === 0) && (c1 * b2 – c2 * b1 === 0)) {
resultDiv.innerHTML = "Infinitely Many Solutions";
} else {
resultDiv.innerHTML = "No Solution";
}
} else {
// Calculate Dx and Dy
var dx = c1 * b2 – c2 * b1;
var dy = a1 * c2 – a2 * c1;
// Calculate x and y
var x = dx / determinant;
var y = dy / determinant;
// Display the result, rounding to a reasonable precision
resultDiv.innerHTML = "x = " + x.toFixed(6) + ", y = " + y.toFixed(6) + "";
}
}