Linear Algebra
1.610 min read

Geometry of Intersecting Lines

The three cases of the trichotomy have precise geometric descriptions. When two lines in R2\mathbb{R}^2 cross at a single point, that point is the unique solution. You can find it algebraically by solving the system.

When the lines are parallel (same slope, different intercept), they never meet. The system is inconsistent and has no solution. This happens algebraically when elimination produces a row 0=c0 = c with c≠0c \neq 0.

When the two equations describe the same line, every point on that line is a solution. The system is under-determined: one equation is redundant. Algebraically, elimination produces a row 0=00 = 0.

The slope of a line a1x1+a2x2=ba_1 x_1 + a_2 x_2 = b is −a1/a2-a_1/a_2. Two lines are parallel exactly when they have the same slope, i.e., the same ratio a1:a2a_1 : a_2.

Formal View

Definition 1.6 — Consistent and Inconsistent Systems
A linear system is consistent if it has at least one solution. It is inconsistent if it has no solution.
Remark 1.6
For two equations in two unknowns, the system (a1b1a2b2)x=(c1c2)\begin{pmatrix} a_1 & b_1 \\ a_2 & b_2 \end{pmatrix} \mathbf{x} = \begin{pmatrix} c_1 \\ c_2 \end{pmatrix} has a unique solution when a1b2−a2b1≠0a_1 b_2 - a_2 b_1 \neq 0 (the lines are not parallel). This expression is the determinant — a concept we will study in depth later.

Why This Matters

Geometric reasoning reveals at a glance whether a system is solvable, saving computation.

  • Computer vision: finding where two camera rays intersect requires solving a 2-variable linear system
  • Game physics: collision detection solves line intersection problems in real time
  • Surveying: triangulation finds a point by intersecting lines from known reference positions

Quiz

Question 1

The system x1+x2=3x_1 + x_2 = 3 and 2x1+2x2=62x_1 + 2x_2 = 6 has:

Common Mistakes

  • Assuming parallel lines (same ratio of coefficients) means no solution — must also check if they are actually different lines.
  • Thinking coincident lines (same equation, scaled) have a "less valid" solution than a unique intersection.