Linear Equations Calculator

Linear Equations Solver (2 Variables)

Enter the coefficients for two linear equations in the form:

ax + by = c

dx + ey = f

Equation 1:

Equation 2:

Understanding Linear Equations

A linear equation is an algebraic equation in which each term has an exponent of one, and the graphing of the equation results in a straight line. The most common form for a linear equation with two variables is Ax + By = C, where A, B, and C are constants, and x and y are the variables.

Systems of Linear Equations

A system of linear equations consists of two or more linear equations with the same variables. The goal is to find values for the variables that satisfy all equations simultaneously. For a system of two linear equations with two variables (x and y), there are three possible outcomes:

1. Unique Solution: The lines intersect at exactly one point. This means there is one specific value for x and one specific value for y that satisfies both equations.
2. No Solution: The lines are parallel and never intersect. This indicates that there are no values for x and y that can satisfy both equations simultaneously.
3. Infinitely Many Solutions: The two equations represent the same line. This means any point on the line is a solution, and there are an infinite number of (x, y) pairs that satisfy both equations.

How This Calculator Works

This calculator solves a system of two linear equations with two variables (x and y) using Cramer's Rule, which is derived from the method of determinants. For a system:

ax + by = c

dx + ey = f

The determinants are calculated as follows:

• Determinant D: D = (a * e) - (b * d)
• Determinant Dx: Dx = (c * e) - (b * f)
• Determinant Dy: Dy = (a * f) - (c * d)

Based on these determinants, the solutions are found:

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

Example Calculation

Let's solve the system:

Equation 1: 2x + 3y = 7

Equation 2: 4x - 2y = 2

Here, a=2, b=3, c=7, d=4, e=-2, f=2.

• D = (2 * -2) - (3 * 4) = -4 - 12 = -16
• Dx = (7 * -2) - (3 * 2) = -14 - 6 = -20
• Dy = (2 * 2) - (7 * 4) = 4 - 28 = -24

Since D = -16 ≠ 0, there is a unique solution:

• x = Dx / D = -20 / -16 = 1.25
• y = Dy / D = -24 / -16 = 1.5

Thus, the solution is x = 1.25 and y = 1.5.