Equations Matrix Calculator

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2×2 Linear Equations Matrix Solver

Enter the coefficients for your system of two linear equations:

Equation 1:
=
Equation 2:
=
Enter values and click "Calculate Solution" to see the results.
function calculateMatrixSolution() { 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 and constants.'; return; } // Calculate the determinant of the coefficient matrix A = [[a1, b1], [a2, b2]] // det(A) = a1*b2 – b1*a2 var determinant = (a1 * b2) – (b1 * a2); if (determinant === 0) { resultDiv.innerHTML = 'The determinant is zero. This system either has no unique solution (parallel lines) or infinitely many solutions (coincident lines).'; return; } // Using Cramer's Rule or inverse matrix method: // x = (b2*c1 – b1*c2) / determinant // y = (a1*c2 – a2*c1) / determinant var x = ((b2 * c1) – (b1 * c2)) / determinant; var y = ((a1 * c2) – (a2 * c1)) / determinant; resultDiv.innerHTML = 'Solution:x = ' + x.toFixed(6) + 'y = ' + y.toFixed(6); }

Understanding the Equations Matrix Calculator

This calculator helps you solve a system of two linear equations with two variables (x and y) using matrix methods. A system of linear equations is a set of two or more linear equations involving the same variables. Finding the solution means finding the values for these variables that satisfy all equations simultaneously.

What is a System of Linear Equations?

A 2×2 system of linear equations typically looks like this:

a1x + b1y = c1
a2x + b2y = c2

Where:

  • x and y are the variables you want to solve for.
  • a1, b1, a2, b2 are the coefficients of the variables.
  • c1 and c2 are the constant terms.

Each equation represents a straight line when plotted on a graph. The solution (x, y) is the point where these two lines intersect.

How Matrices Help Solve Equations

Systems of linear equations can be elegantly represented and solved using matrices. The system above can be written in matrix form as AX = B:

[[a1, b1]
[a2, b2]] * [[x]
[y]] = [[c1]
[c2]]

Here:

  • A is the coefficient matrix: [[a1, b1], [a2, b2]]
  • X is the variable matrix: [[x], [y]]
  • B is the constant matrix: [[c1], [c2]]

To solve for X, we can use the inverse of matrix A (denoted A-1):

X = A-1B

For a 2×2 matrix A = [[a, b], [c, d]], its inverse A-1 is calculated as:

A-1 = (1 / det(A)) * [[d, -b], [-c, a]]

Where det(A) is the determinant of A, calculated as (ad - bc).

This calculator uses these principles (specifically, a method similar to Cramer's Rule, which is derived from matrix inversion) to find the values of x and y.

When is there no unique solution?

If the determinant of the coefficient matrix (a1b2 - b1a2) is zero, the system does not have a unique solution. This means the lines are either parallel (no solution) or coincident (infinitely many solutions).

Example Calculation

Let's use the default values in the calculator:

2x + 3y = 12
4x – 1y = 5

Here, a1=2, b1=3, c1=12 and a2=4, b2=-1, c2=5.

  1. Calculate the Determinant (det):
    det = (a1 * b2) – (b1 * a2)
    det = (2 * -1) – (3 * 4)
    det = -2 – 12 = -14
  2. Calculate x:
    x = ((b2 * c1) – (b1 * c2)) / det
    x = ((-1 * 12) – (3 * 5)) / -14
    x = (-12 – 15) / -14
    x = -27 / -14 ≈ 1.928571
  3. Calculate y:
    y = ((a1 * c2) – (a2 * c1)) / det
    y = ((2 * 5) – (4 * 12)) / -14
    y = (10 – 48) / -14
    y = -38 / -14 ≈ 2.714286

So, the solution is approximately x = 1.928571 and y = 2.714286. You can verify these by plugging them back into the original equations.

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