Solve a system of two linear equations with two variables (x and y) using the substitution or elimination method.
Equation 1:
Equation 2:
Solution:
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 of two linear equations with two variables, typically denoted as 'x' and 'y'. A solution to such a system is an ordered pair (x, y) that satisfies all equations in the system simultaneously.
The general form of a linear equation with two variables is:
Ax + By = C
Where A, B, and C are constants, and at least one of A or B is non-zero.
Methods for Solving Systems of Equations:
Substitution Method:
This method involves solving one of the equations for one variable in terms of the other. Then, substitute this expression into the second equation. This results in a single equation with one variable, which can be solved. Once that variable's value is found, substitute it back into either original equation to find the value of the other variable.
Example Steps:
From Equation 1, solve for y: y = (C1 - A1*x) / B1
Substitute this expression for y into Equation 2: A2*x + B2*((C1 - A1*x) / B1) = C2
Solve the resulting equation for x.
Substitute the found value of x back into the expression for y.
Elimination Method (or Addition Method):
This method involves manipulating one or both equations (by multiplying them by constants) so that the coefficients of one of the variables are opposites. Then, add the two equations together. This eliminates one variable, leaving an equation with only the other variable, which can be solved. Substitute the found value back into one of the original equations to find the value of the eliminated variable.
Example Steps:
Multiply Equation 1 by B2 and Equation 2 by -B1 to eliminate y:
B2*(A1*x + B1*y) = B2*C1 => A1*B2*x + B1*B2*y = C1*B2-B1*(A2*x + B2*y) = -B1*C2 => -A2*B1*x - B1*B2*y = -C2*B1
Add the two modified equations: (A1*B2 - A2*B1)*x = C1*B2 - C2*B1
Solve for x.
Substitute the found value of x back into either original equation to solve for y.
Types of Solutions:
Unique Solution: The lines represented by the equations intersect at exactly one point. This occurs when the slopes of the lines are different. Mathematically, the determinant of the coefficient matrix (A1*B2 - A2*B1) is non-zero.
No Solution: The lines are parallel and distinct, meaning they never intersect. This occurs when the slopes are the same, but the y-intercepts are different. Mathematically, A1*B2 - A2*B1 = 0, but C1*B2 - C2*B1 ≠ 0 (or similar checks involving C and B/A coefficients).
Infinitely Many Solutions: The two equations represent the same line. This occurs when the equations are multiples of each other (same slope and same y-intercept). Mathematically, A1*B2 - A2*B1 = 0 and C1*B2 - C2*B1 = 0 (and other proportional checks).
Use Cases:
Systems of linear equations are fundamental in mathematics and have numerous applications across various fields:
Physics: Analyzing forces, circuits, and motion.
Engineering: Designing structures, optimizing processes, and control systems.
Economics: Modeling market equilibrium, supply and demand.
Computer Graphics: Transformations and projections.
Operations Research: Resource allocation and optimization problems.
Everyday problems: Calculating quantities in mixture problems, determining speeds and distances in travel problems, or solving problems involving multiple unknowns and constraints.
This calculator provides a quick and accurate way to find the solution (or determine the nature of the solution) for a system of two linear equations, helping to visualize and verify results for these diverse applications.
function solveSystem() {
var eq1_coeff_x = parseFloat(document.getElementById("eq1_coeff_x").value);
var eq1_coeff_y = parseFloat(document.getElementById("eq1_coeff_y").value);
var eq1_constant = parseFloat(document.getElementById("eq1_constant").value);
var eq2_coeff_x = parseFloat(document.getElementById("eq2_coeff_x").value);
var eq2_coeff_y = parseFloat(document.getElementById("eq2_coeff_y").value);
var eq2_constant = parseFloat(document.getElementById("eq2_constant").value);
var errorMessageDiv = document.getElementById("error-message");
var resultSectionDiv = document.getElementById("result-section");
var resultOutputDiv = document.getElementById("result-output");
var systemTypeDiv = document.getElementById("system-type");
// Clear previous results and errors
errorMessageDiv.style.display = "none";
resultSectionDiv.style.display = "none";
resultOutputDiv.innerHTML = "";
systemTypeDiv.innerHTML = "";
// Input validation
var inputs = [eq1_coeff_x, eq1_coeff_y, eq1_constant, eq2_coeff_x, eq2_coeff_y, eq2_constant];
for (var i = 0; i < inputs.length; i++) {
if (isNaN(inputs[i])) {
errorMessageDiv.textContent = "Please enter valid numbers for all coefficients and constants.";
errorMessageDiv.style.display = "block";
return;
}
}
// Using Cramer's Rule (Determinants) for a robust solution
// System:
// a1*x + b1*y = c1
// a2*x + b2*y = c2
var a1 = eq1_coeff_x;
var b1 = eq1_coeff_y;
var c1 = eq1_constant;
var a2 = eq2_coeff_x;
var b2 = eq2_coeff_y;
var c2 = eq2_constant;
var determinant = a1 * b2 – a2 * b1;
if (determinant !== 0) {
// Unique solution
var x = (c1 * b2 – c2 * b1) / determinant;
var y = (a1 * c2 – a2 * c1) / determinant;
// Format numbers to avoid excessive decimals, but keep precision
var formattedX = parseFloat(x.toFixed(6));
var formattedY = parseFloat(y.toFixed(6));
resultOutputDiv.innerHTML = "x = " + formattedX + "y = " + formattedY;
systemTypeDiv.innerHTML = "This system has a unique solution.";
resultSectionDiv.style.backgroundColor = "var(–success-green)"; // Default success color
resultSectionDiv.style.display = "block";
} else {
// Determinant is zero, check for no solution or infinite solutions
// Check if the numerators for x and y are also zero
var numeratorX = c1 * b2 – c2 * b1;
var numeratorY = a1 * c2 – a2 * c1;
if (numeratorX === 0 && numeratorY === 0) {
// All coefficients and constants are proportional, likely infinite solutions
// More rigorous check: Ensure at least one variable has non-zero coefficients if the constants are zero
// or check proportionality of all terms
if ((a1 === 0 && b1 === 0 && c1 !== 0) || (a2 === 0 && b2 === 0 && c2 !== 0)) {
resultOutputDiv.innerHTML = "No Solution (Inconsistent equations)";
systemTypeDiv.innerHTML = "This system has no solution.";
resultSectionDiv.style.backgroundColor = "#dc3545"; // Red for error/no solution
} else {
resultOutputDiv.innerHTML = "Infinite Solutions";
systemTypeDiv.innerHTML = "This system has infinitely many solutions.";
resultSectionDiv.style.backgroundColor = "#ffc107"; // Yellow for infinite
}
resultSectionDiv.style.display = "block";
} else {
// Parallel lines, no solution
resultOutputDiv.innerHTML = "No Solution";
systemTypeDiv.innerHTML = "This system has no solution.";
resultSectionDiv.style.backgroundColor = "#dc3545"; // Red for no solution
resultSectionDiv.style.display = "block";
}
}
}