The Linear Equation Solver Calculator is a tool that allows users to solve one or multiple linear equations with one or more variables. Linear equations are algebraic equations in which each term is either a constant or a product of a constant and a variable.
Mathematically, a linear equation can be written as:
a₁x₁ + a₂x₂ + ... + aₙxₙ = b
Where:
a₁, a₂, ..., aₙ are coefficients
x₁, x₂, ..., xₙ are variables
b is a constant
This calculator handles systems of linear equations, providing exact solutions using methods like matrix inversion or Gaussian elimination.
Linear System:
A collection of one or more linear equations involving the same set of variables.
Matrix Representation:
Linear systems can be expressed in matrix form AX=B, where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix.
Gaussian Elimination:
A method for solving linear systems by reducing the augmented matrix to row echelon form.
Cramer’s Rule:
A method using determinants to find solutions of linear systems when the coefficient matrix is invertible.
Single Linear Equation:
x = b / a, for ax = b
System of Linear Equations (Matrix Method):
AX = B ⟹ X = A⁻¹B
Where:
A = Coefficient matrix
X = Variable matrix
B = Constants matrix
A⁻¹ = Inverse of matrix A
Gaussian Elimination Steps:
Form the augmented matrix [A | B]
Apply row operations to reduce it to row echelon form (REF)
Solve for variables using back substitution
Highlight Frame Tip: Place these formulas in a visually distinct frame for better readability and user experience.Problem: Solve the system:
2x + 3y = 8
x - y = 1
Step 1: Represent in matrix form AX=B:
A = [2 3], X = [x], B = [8]
[1 -1] [y] [1]
Step 2: Find the inverse of A:
det(A) = (2)(-1) - (3)(1) = -2 - 3 = -5
A⁻¹ = (1/-5) * [-1 -3] = [0.2 0.6]
[-1 2] [0.2 -0.4]
Step 3: Multiply A⁻¹B:
X = A⁻¹B = [0.2 0.6] [8] = [2.2]
[0.2 -0.4] [1] [1.2]
Result: x = 2.2, y = 1.2