Solve the Linear System of Equations Calculator

Linear System of Equations Solver (2×2)

This calculator helps you solve a system of two linear equations with two variables (x and y). A linear system is typically represented in the form:

Equation 1: a1x + b1y = c1

Equation 2: a2x + b2y = c2

Enter Coefficients and Constants:

Understanding Linear Systems of Equations

A system of linear equations consists of two or more linear equations involving the same variables. The goal is to find values for these variables that satisfy all equations simultaneously. For a 2×2 system, we are looking for a unique pair of (x, y) coordinates that lies on both lines represented by the equations.

Types of Solutions:

  • Unique Solution: This is the most common outcome, where the two lines intersect at exactly one point. The calculator will provide specific values for x and y.
  • No Solution: This occurs when the two lines are parallel and distinct, meaning they never intersect. In this case, the system is inconsistent. The calculator will indicate "No Solution".
  • Infinitely Many Solutions: This happens when the two equations represent the exact same line. Every point on the line is a solution. The calculator will indicate "Infinitely Many Solutions".

How This Calculator Works (Cramer's Rule):

This calculator uses Cramer's Rule, a method that employs determinants to solve systems of linear equations. For a system:

a1x + b1y = c1

a2x + b2y = c2

It calculates three determinants:

  • D (Determinant of the coefficient matrix): D = a1b2 – a2b1
  • Dx (Determinant for x): Dx = c1b2 – c2b1
  • Dy (Determinant for y): Dy = a1c2 – a2c1

If D is not zero, then x = Dx / D and y = Dy / D. If D is zero, the system either has no solution or infinitely many solutions, depending on whether Dx and Dy are also zero.

Example Usage:

Let's solve the system:

2x + 3y = 7

4x – 1y = 1

Input the values:

  • a1 = 2, b1 = 3, c1 = 7
  • a2 = 4, b2 = -1, c2 = 1

Click "Solve System". The calculator will output x = 1 and y = 1.6667 (or 5/3).

Another example for "No Solution":

2x + 4y = 6

1x + 2y = 5

Input: a1=2, b1=4, c1=6, a2=1, b2=2, c2=5. The result will be "No Solution" because the lines are parallel.

function calculateLinearSystem() { var a1 = parseFloat(document.getElementById('a1_coeff').value); var b1 = parseFloat(document.getElementById('b1_coeff').value); var c1 = parseFloat(document.getElementById('c1_const').value); var a2 = parseFloat(document.getElementById('a2_coeff').value); var b2 = parseFloat(document.getElementById('b2_coeff').value); var c2 = parseFloat(document.getElementById('c2_const').value); var resultDiv = document.getElementById('solutionResult'); resultDiv.style.backgroundColor = '#e9f7ef'; // Reset background color resultDiv.style.color = '#155724'; // Reset text color if (isNaN(a1) || isNaN(b1) || isNaN(c1) || isNaN(a2) || isNaN(b2) || isNaN(c2)) { resultDiv.innerHTML = 'Please enter valid numbers for all coefficients and constants.'; resultDiv.style.backgroundColor = '#fff3cd'; resultDiv.style.color = '#856404'; return; } // Calculate the determinant of the coefficient matrix (D) var D = (a1 * b2) – (a2 * b1); // Calculate the determinant for x (Dx) var Dx = (c1 * b2) – (c2 * b1); // Calculate the determinant for y (Dy) var Dy = (a1 * c2) – (a2 * c1); if (D === 0) { if (Dx === 0 && Dy === 0) { resultDiv.innerHTML = 'Result: Infinitely Many Solutions (The equations represent the same line).'; resultDiv.style.backgroundColor = '#d1ecf1'; resultDiv.style.color = '#0c5460'; } else { resultDiv.innerHTML = 'Result: No Solution (The lines are parallel and distinct).'; resultDiv.style.backgroundColor = '#f8d7da'; resultDiv.style.color = '#721c24'; } } else { var x = Dx / D; var y = Dy / D; resultDiv.innerHTML = 'Result:x = ' + x.toFixed(4) + 'y = ' + y.toFixed(4); } }

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