Systems of Linear Equations

The Geometry of Systems

A system of linear equations asks: which values of the unknowns satisfy all equations simultaneously? Geometrically, each equation defines a geometric object—a line in 2D, a plane in 3D. The solution is where these objects intersect.

Consider two equations in two unknowns. Each equation describes a line. The system's solution is the point (or points) lying on both lines.

Three outcomes are possible: the lines intersect at exactly one point (unique solution), the lines are parallel and never meet (no solution), or the lines coincide entirely (infinitely many solutions).

Systems in Three Dimensions

In 3D, each linear equation defines a plane. Three equations give three planes, and we seek their common intersection.

The possibilities multiply: three planes might meet at a single point, along a line, not at all (if two are parallel), or everywhere (if all three are the same plane). Understanding the geometry helps predict what kind of solution to expect.

From Equations to Matrices

A system of linear equations can be written compactly using matrices. The system:

becomes the matrix equation $A\vec{x} = \vec{b}$:

The augmented matrix combines the coefficient matrix and the right-hand side, separating them with a vertical line:

All the information about the system is encoded in this augmented matrix. Solving the system means transforming this matrix into a simpler form.

Row Operations

Three operations transform a system into an equivalent one (same solutions) while simplifying its structure:

• Swap rows: Interchange two equations
• Scale a row: Multiply an equation by a non-zero constant
• Add a multiple of one row to another: The key elimination step

These operations preserve the solution set because they correspond to valid algebraic manipulations of equations. Multiplying both sides of an equation by the same constant, or adding two true equations together, yields another true equation.

Gaussian Elimination

Gaussian elimination systematically applies row operations to transform the augmented matrix into row echelon form: a staircase pattern where each row's leading entry (first non-zero) is to the right of the row above it.

The algorithm proceeds column by column, using the current pivot to eliminate entries below it.

Once in row echelon form, back-substitution solves from the bottom up. The last equation gives one variable directly; substitute upward to find the rest.

Continuing elimination to produce zeros above each pivot as well gives reduced row echelon form (RREF), where each variable's value can be read directly from its row.

Reading the Solution

The final form of the matrix reveals the nature of the solution:

• Unique solution: Every variable is a pivot variable. The matrix reduces to something like the identity on the left.
• No solution: A row becomes [0 0 ... 0 | c] where c ≠ 0. This says 0 = c, a contradiction.
• Infinitely many solutions: Fewer pivots than variables. Free variables can take any value; pivot variables depend on them.

The number of pivots is the rank of the matrix. If rank equals the number of unknowns, and no contradiction appears, the solution is unique.

Connection to Determinants

For a square system (same number of equations as unknowns), the determinant tells you immediately whether a unique solution exists.

If $\det(A) \neq 0$, the columns of $A$ span the full space. Any target vector $\vec{b}$ can be reached by a unique combination of those columns—that combination is the solution.

When the determinant is zero, the columns are linearly dependent—they collapse onto a lower-dimensional subspace. The system either has no solution (if $\vec{b}$ is not in that subspace) or infinitely many (if $\vec{b}$ happens to lie in it).

Key Takeaways

• A system of linear equations asks where geometric objects (lines, planes) intersect
• Systems can have one solution, no solution, or infinitely many solutions
• The augmented matrix compactly represents the system; row operations transform it while preserving solutions
• Gaussian elimination reduces the matrix to row echelon form, enabling back-substitution
• The determinant of a square system's coefficient matrix determines whether a unique solution exists
• $\det(A) \neq 0$ means unique solution; $\det(A) = 0$ means either no solution or infinitely many