Solve a system of two linear equations using the elimination method.
Equation 1: ax + by = c
Equation 2: dx + ey = f
Enter the coefficients and constants for both equations to find the solution (x, y).
Understanding the Elimination Method for Systems of Equations
Solving systems of linear equations is a fundamental skill in algebra, with applications ranging from economics and engineering to computer graphics and physics. The elimination method, also known as the addition method, provides an elegant way to find the unique solution (if one exists) for a system of two linear equations with two variables.
A system of two linear equations typically looks like this:
Equation 1: ax + by = c
Equation 2: dx + ey = f
Here, a, b, c, d, e, f are known constants, and x and y are the variables we aim to solve for. The goal of the elimination method is to manipulate one or both equations (by multiplying them by constants) such that the coefficients of either the x or y terms are opposites. When the modified equations are added together, one of the variables will be eliminated, allowing us to solve for the remaining variable.
Steps of the Elimination Method:
Align Equations: Ensure both equations are written in the standard form Ax + By = C, with variables aligned vertically.
Prepare for Elimination:
Option A (Eliminate y): Multiply Equation 1 by e and Equation 2 by -b. This makes the coefficients of y become be and -be.
Option B (Eliminate x): Multiply Equation 1 by d and Equation 2 by -a. This makes the coefficients of x become ad and -ad.
Alternatively, if coefficients are already the same or opposites, you might only need to multiply one equation or just add/subtract directly.
Add or Subtract Equations: Add the two modified equations. One variable should cancel out.
Solve for the Remaining Variable: Solve the resulting single-variable equation for the variable that was not eliminated.
Substitute Back: Substitute the value found in Step 4 into either of the original equations to solve for the other variable.
Check the Solution: Substitute the values of both variables (x, y) into both original equations to verify that they hold true.
The Math Behind the Calculator:
Let the system be:
ax + by = c (1)
dx + ey = f (2)
To eliminate y:
Multiply (1) by e: aex + bey = ce Multiply (2) by -b: -bdx - bey = -bf Add these: (ae - bd)x = ce - bf So, x = (ce - bf) / (ae - bd)
To eliminate x:
Multiply (1) by d: adx + bdy = cd Multiply (2) by -a: -adx - aey = -af Add these: (bd - ae)y = cd - af So, y = (cd - af) / (bd - ae)
The denominator (ae - bd) is crucial. If ae - bd = 0:
If ce - bf is also 0, there are infinitely many solutions (the lines are coincident).
If ce - bf is not 0, there is no solution (the lines are parallel and distinct).
If ae - bd ≠ 0, there is a unique solution.
Use Cases:
Finding Intersection Points: Determine where two lines on a graph intersect.
Resource Allocation: Optimize the use of limited resources in business or manufacturing.
Mixture Problems: Calculate the amounts of different substances needed to achieve a specific mixture concentration.
Physics and Engineering: Solve circuit analysis problems, analyze forces, or model motion.
function calculateElimination() {
var a = parseFloat(document.getElementById("a").value);
var b = parseFloat(document.getElementById("b").value);
var c = parseFloat(document.getElementById("c").value);
var d = parseFloat(document.getElementById("d").value);
var e = parseFloat(document.getElementById("e").value);
var f = parseFloat(document.getElementById("f").value);
var resultDiv = document.getElementById("result");
resultDiv.className = ""; // Reset classes
// Check if inputs are valid numbers
if (isNaN(a) || isNaN(b) || isNaN(c) || isNaN(d) || isNaN(e) || isNaN(f)) {
resultDiv.innerHTML = "Please enter valid numbers for all coefficients and constants.";
resultDiv.classList.add("error");
return;
}
var denominator = (a * e) – (b * d);
if (denominator === 0) {
// Check for no solution or infinite solutions
var numeratorX = (c * e) – (b * f);
var numeratorY = (a * f) – (c * d); // Note: Using a slightly different form for y numerator for consistency check
if (numeratorX === 0 && numeratorY === 0) {
// Check if the equations are identical or multiples
// If a/d = b/e = c/f (and denominators aren't zero), then infinite solutions
// We already know ae – bd = 0, so a*e = b*d
// If a=0, b=0, c=0, then infinite (trivial case)
// If d=0, e=0, f=0, then infinite (trivial case)
// If the non-zero coefficients are proportional:
var ratiosMatch = true;
if (d !== 0 && a / d !== e / b) ratiosMatch = false; // check a/d vs b/e
if (b !== 0 && a / b !== d / e) ratiosMatch = false; // check a/b vs d/e
if (e !== 0 && b / e !== a / d) ratiosMatch = false; // check b/e vs a/d
if (a !== 0 && d / a !== e / b) ratiosMatch = false; // check d/a vs e/b
if (c !== 0 && f !== 0 && (a/d !== c/f || b/e !== c/f)) { // Check constants proportionality too
// Need to be careful with divisions by zero
// A safer check is cross-multiplication: ae = bd, af = cd, bf = ce
if (! ( (a*e === b*d) && (a*f === c*d) && (b*f === c*e) ) ) {
ratiosMatch = false;
}
} else if (c === 0 && f !== 0 || c !== 0 && f === 0) {
ratiosMatch = false; // If one constant is zero and other isn't, but coeffs are proportional
}
// A simpler check: if ae-bd=0 AND ce-bf=0, AND af-cd=0, then infinite.
// Otherwise, no solution.
if ( (c*e – b*f === 0) && (a*f – c*d === 0) ) {
resultDiv.innerHTML = "Infinite solutions (the equations represent the same line).";
resultDiv.classList.add("success");
} else {
resultDiv.innerHTML = "No solution (the lines are parallel and distinct).";
resultDiv.classList.add("error");
}
} else {
resultDiv.innerHTML = "No solution (the lines are parallel and distinct).";
resultDiv.classList.add("error");
}
} else {
var x = ((c * e) – (b * f)) / denominator;
var y = ((a * f) – (c * d)) / denominator;
// Check if values are extremely close to zero and should be treated as zero
var tolerance = 1e-10; // Small tolerance for floating point comparisons
if (Math.abs(x) < tolerance) x = 0;
if (Math.abs(y) < tolerance) y = 0;
resultDiv.innerHTML = `Solution: x = ${x.toFixed(5)}, y = ${y.toFixed(5)}`;
resultDiv.classList.add("success");
}
}