Solve systems of two linear equations with two variables (x and y).
Solutions will appear here.
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 systems with two variables, typically denoted as 'x' and 'y'. Each linear equation represents a straight line when plotted on a graph. The solution to a system of linear equations is the set of values for the variables that satisfy all equations simultaneously. Geometrically, this corresponds to the point(s) where all the lines in the system intersect.
The General Form
A system of two linear equations with two variables can be represented in the general form:
a₁x + b₁y = c₁
a₂x + b₂y = c₂
Where a₁, b₁, c₁, a₂, b₂, and c₂ are coefficients and constants. Our calculator takes these coefficients and constants as input to find the values of x and y that make both equations true.
Methods for Solving
There are several methods to solve systems of linear equations, including:
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.
Graphical Method: Graph both lines and find the point of intersection.
Matrix Methods (e.g., Cramer's Rule): Use determinants or matrix inverses to solve.
This calculator utilizes a method derived from Cramer's Rule or the elimination method, which involves calculating determinants or specific combinations of the coefficients.
Step 3: Calculate Dy Dy = (2)(6) - (4)(7) = 12 - 28 = -16
Step 4: Find x and y x = Dₓ / D = -32 / -16 = 2 y = Dy / D = -16 / -16 = 1
The solution is x = 2, y = 1.
function calculateSystem() {
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
resultDiv.classList.remove("error");
// Check if inputs are valid numbers
if (isNaN(a1) || isNaN(b1) || isNaN(c1) || isNaN(a2) || isNaN(b2) || isNaN(c2)) {
resultDiv.innerHTML = "Error: Please enter valid numbers for all coefficients and constants.";
resultDiv.classList.add("error");
return;
}
// Calculate the determinant of the coefficient matrix
var determinant = a1 * b2 – a2 * b1;
// Check for special cases
if (determinant === 0) {
// Check if the system is consistent (infinitely many solutions or no solution)
// We can check this by seeing if the equations are proportional
// If a1/a2 = b1/b2 = c1/c2 (handling division by zero)
var isProportional = false;
if (a1 === 0 && a2 === 0 && b1 === 0 && b2 === 0) { // Both equations are 0 = c1/c2
if (c1 === c2) isProportional = true;
} else if (a1 === 0 && a2 === 0) { // equations are b1*y = c1 and b2*y = c2
if (b1 !== 0 && b2 !== 0 && (c1 / b1) === (c2 / b2)) isProportional = true;
} else if (b1 === 0 && b2 === 0) { // equations are a1*x = c1 and a2*x = c2
if (a1 !== 0 && a2 !== 0 && (c1 / a1) === (c2 / a2)) isProportional = true;
} else if (a1 !== 0 && a2 !== 0 && b1 !== 0 && b2 !== 0) { // general case check proportionality
if (Math.abs((a1 / a2) – (b1 / b2)) < 1e-9 && Math.abs((b1 / b2) – (c1 / c2)) < 1e-9) {
isProportional = true;
}
} else { // Mixed cases like a1x = c1 and a2x + b2y = c2 etc.
// A more robust check for proportionality when coefficients are zero
// If determinant is 0, the lines are parallel or coincident.
// To distinguish, check if a point on one line satisfies the other.
// Or check if (a1*c2 – a2*c1) or (b1*c2 – b2*c1) are zero.
var detX_check = a1 * c2 – a2 * c1;
var detY_check = b1 * c2 – b2 * c1;
if (Math.abs(detX_check) < 1e-9 && Math.abs(detY_check) < 1e-9) {
isProportional = true;
}
}
if (isProportional) {
resultDiv.innerHTML = "The system has infinitely many solutions.";
} else {
resultDiv.innerHTML = "The system has no solution (parallel lines).";
}
resultDiv.classList.add("error");
} else {
// Calculate determinants for x and y
var detX = c1 * b2 – c2 * b1;
var detY = a1 * c2 – a2 * c1;
// Calculate the solution
var x = detX / determinant;
var y = detY / determinant;
// Display the result, rounding to a reasonable precision
resultDiv.innerHTML = `Solution:x = ${x.toFixed(4)}y = ${y.toFixed(4)}`;
}
}