Linear System Calculator

by

Linear System Solver (2×2)

Enter the coefficients for your system of two linear equations:

Equation 1: a₁x + b₁y = c₁
Equation 2: a₂x + b₂y = c₂
function calculateLinearSystem() { 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'); if (isNaN(a1) || isNaN(b1) || isNaN(c1) || isNaN(a2) || isNaN(b2) || isNaN(c2)) { resultDiv.innerHTML = 'Please enter valid numbers for all coefficients.'; resultDiv.style.backgroundColor = '#f8d7da'; resultDiv.style.borderColor = '#f5c6cb'; resultDiv.style.color = '#721c24'; return; } // Calculate the determinant D var D = (a1 * b2) – (a2 * b1); // Calculate determinants Dx and Dy var Dx = (c1 * b2) – (c2 * b1); var Dy = (a1 * c2) – (a2 * c1); if (D === 0) { if (Dx === 0 && Dy === 0) { resultDiv.innerHTML = 'The system has infinitely many solutions (the lines are coincident).'; resultDiv.style.backgroundColor = '#fff3cd'; resultDiv.style.borderColor = '#ffeeba'; resultDiv.style.color = '#856404'; } else { resultDiv.innerHTML = 'The system has no solution (the lines are parallel and distinct).'; resultDiv.style.backgroundColor = '#f8d7da'; resultDiv.style.borderColor = '#f5c6cb'; resultDiv.style.color = '#721c24'; } } else { var x = Dx / D; var y = Dy / D; resultDiv.innerHTML = 'Solution:x = ' + x.toFixed(4) + 'y = ' + y.toFixed(4) + ''; resultDiv.style.backgroundColor = '#e9f7ef'; resultDiv.style.borderColor = '#d4edda'; resultDiv.style.color = '#155724'; } }

Understanding Linear Systems and How to Solve Them

A linear system of equations is a set of two or more linear equations involving the same variables. In a 2×2 linear system, we deal with two equations and two variables, typically denoted as 'x' and 'y'. The goal is to find the values of 'x' and 'y' that satisfy both equations simultaneously.

The Standard Form of a 2×2 Linear System

A 2×2 linear system is generally written in the following form:

Equation 1: a₁x + b₁y = c₁
Equation 2: a₂x + b₂y = c₂

Here, a₁, b₁, c₁, a₂, b₂, and c₂ are known coefficients and constants, while 'x' and 'y' are the variables we aim to solve for.

How the Calculator Works: Cramer's Rule

This calculator uses Cramer's Rule, a method that uses determinants to solve systems of linear equations. For a 2×2 system, the steps are as follows:

  1. Calculate the main determinant (D):

    D = (a₁ * b₂) – (a₂ * b₁)

  2. Calculate the determinant for x (Dx):

    Dx = (c₁ * b₂) – (c₂ * b₁)

  3. Calculate the determinant for y (Dy):

    Dy = (a₁ * c₂) – (a₂ * c₁)

  4. Find x and y:

    x = Dx / D
    y = Dy / D

Types of Solutions

The value of the main determinant (D) determines the nature of the solution:

  • Unique Solution (D ≠ 0): If D is not zero, there is exactly one unique pair of (x, y) values that satisfies both equations. Geometrically, this means the two lines intersect at a single point.
  • No Solution (D = 0, but Dx ≠ 0 or Dy ≠ 0): If D is zero, but at least one of Dx or Dy is not zero, the system has no solution. This indicates that the lines are parallel and distinct, meaning they never intersect.
  • Infinitely Many Solutions (D = 0, Dx = 0, and Dy = 0): If D, Dx, and Dy are all zero, the system has infinitely many solutions. This occurs when the two equations represent the same line (they are coincident).

Example Calculation

Let's solve the following system using the calculator:

2x + 3y = 7
4x – 2y = 6

Here, the coefficients are:

  • a₁ = 2, b₁ = 3, c₁ = 7
  • a₂ = 4, b₂ = -2, c₂ = 6

Using the formulas:

  • D = (2 * -2) – (4 * 3) = -4 – 12 = -16
  • Dx = (7 * -2) – (6 * 3) = -14 – 18 = -32
  • Dy = (2 * 6) – (4 * 7) = 12 – 28 = -16

Since D is not zero, we have a unique solution:

  • x = Dx / D = -32 / -16 = 2
  • y = Dy / D = -16 / -16 = 1

Thus, the solution to this system is x = 2 and y = 1. You can input these values into the calculator to verify the result.

Leave a Comment