Systems of Equations Calculator

Systems of Equations Calculator

This calculator helps you solve a system of two linear equations with two variables (x and y) in the standard form:

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

Enter the coefficients and constants for your two equations below.

Equation 1:

Equation 2:

function calculateSystem() { 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"); resultDiv.style.backgroundColor = '#d4edda'; // Default success color resultDiv.style.color = '#155724'; 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 = '#f8d7da'; // Error color resultDiv.style.color = '#721c24'; return; } // Calculate 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 (dependent system)."; resultDiv.style.backgroundColor = '#fff3cd'; // Warning color resultDiv.style.color = '#856404'; } else { resultDiv.innerHTML = "The system has no solution (inconsistent system)."; resultDiv.style.backgroundColor = '#f8d7da'; // Error color resultDiv.style.color = '#721c24'; } } else { var x = Dx / D; var y = Dy / D; resultDiv.innerHTML = "Solution:x = " + x.toFixed(4) + "y = " + y.toFixed(4); } }

Understanding Systems of Linear Equations

A system of linear equations consists of two or more linear equations with the same set of variables. For a system of two equations with two variables (commonly 'x' and 'y'), we are looking for a pair of values (x, y) that satisfies both equations simultaneously. Graphically, this solution represents the point where the lines represented by each equation intersect.

Why Solve Systems of Equations?

Systems of equations are fundamental in mathematics, science, engineering, economics, and many other fields. They are used to model real-world problems where multiple conditions or relationships need to be satisfied at once. For example:

• Physics: Calculating forces, velocities, or trajectories.
• Economics: Determining equilibrium prices and quantities in supply and demand models.
• Engineering: Analyzing electrical circuits or structural loads.
• Chemistry: Balancing chemical equations.

How This Calculator Works (Cramer's Rule)

This calculator solves systems of two linear equations using a method called Cramer's Rule. For a system in the form:

a₁x + b₁y = c₁
a₂x + b₂y = c₂

Cramer's Rule involves calculating three determinants:

1. Determinant D: D = a₁b₂ – a₂b₁
2. Determinant D_x: D_x = c₁b₂ – c₂b₁
3. Determinant D_y: D_y = a₁c₂ – a₂c₁

Once these determinants are found, the solutions for x and y are:

x = D_x / D
y = D_y / D

Possible Outcomes:

• Unique Solution (D ≠ 0): The lines intersect at a single point, yielding a unique (x, y) pair.
• No Solution (D = 0, but D_x ≠ 0 or D_y ≠ 0): The lines are parallel and distinct, meaning they never intersect.
• Infinitely Many Solutions (D = 0, D_x = 0, and D_y = 0): The lines are identical (coincident), meaning every point on the line is a solution.

Example Calculation:

Let's solve the following system:

2x + 3y = 12
5x – 2y = 11

Here, the coefficients are:

• a₁ = 2, b₁ = 3, c₁ = 12
• a₂ = 5, b₂ = -2, c₂ = 11

Using Cramer's Rule:

• D = (2)(-2) – (5)(3) = -4 – 15 = -19
• D_x = (12)(-2) – (11)(3) = -24 – 33 = -57
• D_y = (2)(11) – (5)(12) = 22 – 60 = -38

Now, find x and y:

• x = D_x / D = -57 / -19 = 3
• y = D_y / D = -38 / -19 = 2

The solution is x = 3 and y = 2. You can verify this by plugging these values back into the original equations.