Online Calculator Algebra

System of Two Linear Equations Solver

Enter the coefficients and constants for your system of two linear equations:
Equation 1: ax + by = c
Equation 2: dx + ey = f

function calculateSystem() { var a = parseFloat(document.getElementById("coeffA").value); var b = parseFloat(document.getElementById("coeffB").value); var c = parseFloat(document.getElementById("constantC").value); var d = parseFloat(document.getElementById("coeffD").value); var e = parseFloat(document.getElementById("coeffE").value); var f = parseFloat(document.getElementById("constantF").value); var resultDiv = document.getElementById("resultOutput"); if (isNaN(a) || isNaN(b) || isNaN(c) || isNaN(d) || isNaN(e) || isNaN(f)) { resultDiv.innerHTML = "Please enter valid numbers for all coefficients and constants."; return; } // Calculate determinants using Cramer's Rule var determinant = (a * e) – (b * d); var determinantX = (c * e) – (b * f); var determinantY = (a * f) – (c * d); if (determinant === 0) { if (determinantX === 0 && determinantY === 0) { resultDiv.innerHTML = "The system has infinitely many solutions (dependent system)."; } else { resultDiv.innerHTML = "The system has no solution (inconsistent system)."; } } else { var x = determinantX / determinant; var y = determinantY / determinant; resultDiv.innerHTML = "Solution:x = " + x.toFixed(4) + "y = " + y.toFixed(4); } }

Understanding Systems of Linear Equations

In algebra, a system of linear equations is a collection of two or more linear equations involving the same set of variables. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable (to the first power). For example, 2x + 3y = 7 is a linear equation.

When we have a system of two linear equations with two variables (commonly x and y), we are looking for a pair of values (x, y) that satisfies BOTH equations simultaneously. Geometrically, each linear equation represents a straight line. The solution to the system is the point where these two lines intersect.

Forms of Solutions:

• One Solution: The most common case, where the two lines intersect at a single unique point. Our calculator will provide these specific x and y values.
• No Solution: If the two lines are parallel and distinct, they will never intersect. In this case, there is no (x, y) pair that satisfies both equations.
• Infinitely Many Solutions: If the two equations represent the exact same line (one is a multiple of the other), then every point on that line is a solution.

How Our Calculator Works (Cramer's Rule):

Our calculator uses a method called Cramer's Rule to solve systems of two linear equations. This rule is particularly useful for systems with a small number of equations and variables, as it provides a direct formula for the solution using determinants.

Consider a system of two linear equations in the form:
ax + by = c (Equation 1)
dx + ey = f (Equation 2)

Cramer's Rule involves calculating three determinants:

1. Determinant of the Coefficient Matrix (D): This is calculated from the coefficients of x and y.
   D = (a * e) - (b * d)
2. Determinant for x (Dx): This is found by replacing the x-coefficients in D with the constants c and f.
   Dx = (c * e) - (b * f)
3. Determinant for y (Dy): This is found by replacing the y-coefficients in D with the constants c and f.
   Dy = (a * f) - (c * d)

Once these determinants are calculated:

• If D ≠ 0, then there is a unique solution:
  x = Dx / D
y = Dy / D
• If D = 0 and Dx = 0 and Dy = 0, there are infinitely many solutions.
• If D = 0 but either Dx ≠ 0 or Dy ≠ 0 (or both), there is no solution.

Example Usage:

Let's solve the system:
2x + 3y = 7
4x - 1y = 1

Here, the coefficients and constants are:

• a = 2
• b = 3
• c = 7
• d = 4
• e = -1
• f = 1

Using the calculator with these values will yield the solution for x and y.

This calculator is a handy tool for quickly verifying solutions or solving systems of linear equations without manual calculation, making it a great resource for students and professionals alike.