Sistem Persamaan Linier 3 Variabel

Introduction

• A system of linear equations with three variables is a set of three linear equations that involve three variables.
• The goal is to find the values of the three variables that satisfy all three equations.

General Form

• The general form of a system of linear equations with three variables is:

a1x + b1y + c1z = d1
a2x + b2y + c2z = d2
a3x + b3y + c3z = d3

• Where a1, b1, c1, d1, a2, b2, c2, d2, a3, b3, c3, and d3 are constants, and x, y, and z are the variables.

Solution Methods

• There are several methods to solve a system of linear equations with three variables, including:
  1. Substitution Method: Solve one equation for one variable, and then substitute that expression into the other two equations.
  2. Elimination Method: Add or subtract equations to eliminate one variable, and then solve for the remaining variables.
  3. Matrix Method: Use matrices to represent the system of equations, and then solve using matrix operations.

Consistency and Independence

• A system of linear equations with three variables can be:
  • Consistent: Has a unique solution.
  • Inconsistent: Has no solution.
  • Dependent: Has infinitely many solutions.
• The consistency and independence of the system can be determined by finding the rank of the coefficient matrix.

Graphical Representation

• A system of linear equations with three variables can be represented graphically in a 3D coordinate system.
• Each equation represents a plane in 3D space, and the solution to the system is the point of intersection of the three planes.
• If the system is inconsistent, the planes do not intersect. If the system is dependent, the planes intersect in a line or a plane.