Enter the coefficients for your system of linear equations.
Understanding the Solve by Elimination Method
The Solve by Elimination method is a powerful technique for solving systems of linear equations. A system of linear equations is a set of two or more equations, each involving two or more variables. The goal is to find the values of these variables that satisfy all equations simultaneously. The elimination method works by strategically manipulating the equations so that one of the variables cancels out (is "eliminated") when the equations are added or subtracted, leaving an equation with only one variable that can be easily solved.
How the Elimination Method Works (for two variables)
Consider a system of two linear equations with two variables, x and y:
Equation 1: a₁x + b₁y = c₁
Equation 2: a₂x + b₂y = c₂
The steps involved are:
Align the Equations: Ensure both equations are in the standard form Ax + By = C, with variables aligned vertically.
Make Coefficients Opposites: The key is to make the coefficients of either the 'x' or 'y' variable opposites. You can achieve this by multiplying one or both equations by a suitable constant. For example, if you want to eliminate 'x':
If a₁ and a₂ are not already opposites (e.g., 2 and -2), multiply Equation 1 by -a₂ and Equation 2 by a₁. This will make the 'x' coefficients -a₁a₂ and a₁a₂.
Alternatively, if you want to eliminate 'y', make b₁ and b₂ opposites.
Add or Subtract Equations: Once the coefficients for one variable are opposites, add the two modified equations together. The variable with opposite coefficients will cancel out (sum to zero).
Solve for the Remaining Variable: You will be left with a simple linear equation in one variable. Solve this equation.
Substitute Back: Substitute the value of the variable you just found into either of the original equations.
Solve for the Second Variable: Solve the resulting equation for the second variable.
Check Your Solution: Substitute both found values back into both original equations to verify that they hold true.
Calculator Logic Explained
This calculator implements the elimination method for a system of two linear equations:
The calculator first determines which variable to eliminate. It calculates the factor needed to make the 'x' coefficients opposites and the factor needed to make the 'y' coefficients opposites.
The calculator checks for division by zero (parallel or identical lines) and calculates x and y using these derived formulas.
Use Cases
Algebraic Problem Solving: Solving word problems that can be translated into a system of linear equations (e.g., problems involving rates, mixtures, or quantities).
Engineering and Physics: Analyzing circuits, determining forces, or solving mechanics problems where multiple constraints lead to linear systems.
Economics: Modeling supply and demand curves, cost analysis, and break-even points.
Computer Graphics: Solving for transformations or intersections.
Resource Allocation: Determining optimal distribution of resources based on constraints.
function calculateSolution() {
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 resultDiv = document.getElementById("result");
resultDiv.innerHTML = "; // Clear previous result
// Check if inputs are valid numbers
if (isNaN(eq1_coeff_x) || isNaN(eq1_coeff_y) || isNaN(eq1_constant) ||
isNaN(eq2_coeff_x) || isNaN(eq2_coeff_y) || isNaN(eq2_constant)) {
resultDiv.innerHTML = 'Error: Please enter valid numbers for all coefficients and constants.';
return;
}
// Calculate determinant of the coefficient matrix
var determinant = (eq1_coeff_x * eq2_coeff_y) – (eq2_coeff_x * eq1_coeff_y);
if (determinant === 0) {
// Check if the equations are dependent (infinitely many solutions)
// This happens if (A1*C2 – A2*C1 == 0) AND (B1*C2 – B2*C1 == 0)
// or more simply, if C1/A1 = C2/A2 = B1/A1 = B2/A2 (handle division by zero carefully)
// A simpler check for dependency when determinant is 0:
// Check if ratios of corresponding coefficients are equal
var ratio_x = (eq2_coeff_x === 0) ? (eq1_coeff_x === 0 ? 1 : Infinity) : eq1_coeff_x / eq2_coeff_x;
var ratio_y = (eq2_coeff_y === 0) ? (eq1_coeff_y === 0 ? 1 : Infinity) : eq1_coeff_y / eq2_coeff_y;
var ratio_c = (eq2_constant === 0) ? (eq1_constant === 0 ? 1 : Infinity) : eq1_constant / eq2_constant;
// Handle cases where denominators are zero for ratios
if (eq2_coeff_x === 0 && eq1_coeff_x !== 0) ratio_x = Infinity;
if (eq2_coeff_x === 0 && eq1_coeff_x === 0) ratio_x = 1; // Both are zero
if (eq2_coeff_y === 0 && eq1_coeff_y !== 0) ratio_y = Infinity;
if (eq2_coeff_y === 0 && eq1_coeff_y === 0) ratio_y = 1; // Both are zero
if (eq2_constant === 0 && eq1_constant !== 0) ratio_c = Infinity;
if (eq2_constant === 0 && eq1_constant === 0) ratio_c = 1; // Both are zero
if (ratio_x === ratio_y && ratio_y === ratio_c && ratio_x !== Infinity) {
resultDiv.innerHTML = 'The system has infinitely many solutions (the equations are dependent).';
} else {
resultDiv.innerHTML = 'The system has no solution (the lines are parallel).';
}
} else {
// Calculate solutions using Cramer's Rule (which is derived from elimination)
var x_numerator = (eq1_constant * eq2_coeff_y) – (eq2_constant * eq1_coeff_y);
var y_numerator = (eq1_coeff_x * eq2_constant) – (eq2_coeff_x * eq1_constant);
var x = x_numerator / determinant;
var y = y_numerator / determinant;
resultDiv.innerHTML = 'Solution: x = ' + x.toFixed(4) + ', y = ' + y.toFixed(4) + ";
}
}