Systems of Linear Equations

Systems & Geometry

A linear equation uses only sums and scalar multiples, like $2x + 3y = 6$. A system requires several to hold simultaneously, written compactly as $A\mathbf{x} = \mathbf{b}$ where $A$ is the matrix, $\mathbf{x}$ is the variables, and $\mathbf{b}$ is the constants.

Geometrically, each equation in two variables is a line. A solution is a point on all lines at once, leading to three geometric cases:

Common pitfall: Assuming every system has a unique solution. Always check for parallel lines or redundant equations.

Gaussian Elimination

The systematic method is Gaussian elimination: apply row operations to the augmented matrix $[A \mid \mathbf{b}]$ to reach row-echelon form, then back-substitute.

Allowed row operations (which never change the solution set):

• Swap two rows
• Scale a row by a non-zero constant
• Add a multiple of one row to another