System of Linear Equations Calculator
A system of linear equations is a collection of two or more linear equations involving the same set of variables. This calculator helps you solve a system of two linear equations with two variables (x and y) in the standard form:
a2x + b2y = c2
These systems are fundamental in mathematics, science, engineering, and economics for modeling real-world problems where multiple conditions or relationships must be satisfied simultaneously. For example, they can be used to find the intersection point of two lines, determine optimal resource allocation, or analyze electrical circuits.
Enter the coefficients (a, b) and constants (c) for each equation below. The calculator will determine the values of x and y that satisfy both equations, or indicate if there's no unique solution.
Enter Equation Coefficients and Constants:
Enter values and click "Solve System" to see the solution.
Understanding the Results
When you solve a system of two linear equations, there are three possible outcomes:
- Unique Solution: The two lines intersect at exactly one point. This calculator will provide specific values for x and y.
- No Solution: The two lines are parallel and never intersect. This is an "inconsistent system." The calculator will indicate "No solution."
- Infinitely Many Solutions: The two equations represent the same line (they are coincident). This is a "dependent system." The calculator will indicate "Infinitely many solutions."
This calculator uses Cramer's Rule, a method that involves determinants, to find the solution for x and y.
Example Scenarios:
Example 1: Unique Solution
Consider the system:
3x – y = 3
Here, a1=2, b1=1, c1=7, a2=3, b2=-1, c2=3. Inputting these values will yield x=2 and y=3.
Example 2: No Solution (Parallel Lines)
Consider the system:
4x + 2y = 12
Here, a1=2, b1=1, c1=5, a2=4, b2=2, c2=12. Notice that the second equation is 2 times the first on the left side (4x+2y), but the constant on the right side (12) is not 2 times 5. This indicates parallel lines. The calculator will show "No solution."
Example 3: Infinitely Many Solutions (Coincident Lines)
Consider the system:
2x + 4y = 8
Here, a1=1, b1=2, c1=4, a2=2, b2=4, c2=8. The second equation is simply 2 times the first equation. Both equations represent the same line. The calculator will show "Infinitely many solutions."
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('calculationResult'); resultDiv.innerHTML = "; // Clear previous results if (isNaN(a1) || isNaN(b1) || isNaN(c1) || isNaN(a2) || isNaN(b2) || isNaN(c2)) { resultDiv.innerHTML = 'Please enter valid numbers for all coefficients and constants.'; 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 (Dependent System)'; } else { resultDiv.innerHTML = 'Result: No solution (Inconsistent System)'; } } else { var x = Dx / D; var y = Dy / D; resultDiv.innerHTML = 'Result:' + 'x = ' + x.toFixed(4) + " + 'y = ' + y.toFixed(4) + "; } }