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Gaussian elimination solves systems of linear equations by using one equation to eliminate a variable from the others. The method simplifies the system into an upper triangular form through elementary row operations like multiplying and subtracting equations. Once in triangular form, back-substitution determines variable values starting from the bottom equation and working upward to find the complete solution.

To eliminate a variable, choose which variable to remove first. Multiply one equation by a constant, then subtract it from another equation containing that variable. This creates a new equation without the chosen variable. Repeat this process for each variable across all equations to progressively build the triangular form.

Back-substitution is used after the system is converted to upper triangular form. Starting with the bottom equation, solve for the last variable. Substitute that value into the equation above it to find the next variable. Continue this process upward through all equations until all variable values are determined.

A system of three equations can have three solution types: a unique solution when the three planes intersect at a single point; infinitely many solutions when planes intersect along a line or overlap entirely; or no solution when planes are parallel or do not intersect at a common point. The geometric configuration determines which outcome occurs.

Elementary row operations are manipulations used to simplify a system without changing its solution. These include swapping rows to place non-zero coefficients in leading positions, scaling rows by multiplying by constants, and eliminating variables by subtracting linear combinations of rows. These operations transform the system into upper triangular form.

Gaussian elimination models complex scenarios with multiple unknowns and constraints. For example, factory production problems use variables for run days and equations for output constraints. By solving the resulting system, you find how many days each factory must operate to meet production goals, demonstrating how linear equations represent real resource allocation and planning decisions.

Upper triangular form simplifies solving because each equation contains progressively fewer variables. The bottom equation has only one variable, making it solvable directly. Each equation above contains one additional variable, allowing systematic back-substitution. This structured arrangement makes finding all variable values efficient and organized.