Visighta

Section 1.1 — Visualizing Systems of Two Linear Equations

In this section we build geometric intuition for $2 \times 2$ linear systems. We will see that:

  • A single linear equation in two variables describes a line.
  • A system of two equations corresponds to two lines, and the intersection tells us the solution.
  • Depending on the system, there can be one, zero, or infinitely many solutions.
from linear_algebra_course.chapter1.section1.scenes import *
from drawsvg_renderer import DrawSVGRenderer

renderer = DrawSVGRenderer(
    width=800,
    height=600,
    frame_width=16.0,
    background_color="#000000",
    output_mode='jupyter',
    fps=30,
    show_progress=False
)

Scene 1 — One Equation, One Line

An equation in two unknowns, such as $x + y = 4$, describes all points $(x, y)$ that satisfy it.
Geometrically these points form a line.

Below we plot the line and mark several points that satisfy the equation (yellow) versus points that do not (red).

renderer.display_all(Scene1_OneEquationOneLine(), display=False)
renderer._finalize_interactivity()
renderer.display_inline()
Cell visualization output

Scene 2 — Two Equations, Two Lines, One Intersection

Now consider a system of two equations:

$$ \begin{cases} x + y = 4 \ x - y = 0 \end{cases} $$

Each equation gives a line. The intersection point $(2, 2)$ is the unique solution.

renderer.display_all(Scene2_TwoLineIntersection(), display=False)
renderer._finalize_interactivity()
renderer.display_inline()
Cell visualization output

Scene 3 — Algebra Meets Geometry

We can solve the system algebraically while tracking the solution on the graph.

From the second equation $x - y = 0$ we get $x = y$.
Substituting into the first: $y + y = 4 \Rightarrow y = 2$, so $x = 2$.

The algebraic steps narrow attention to the intersection point $(2, 2)$ on the graph.

renderer.display_all(Scene3_AlgebraMeetsGeometry(), display=False)
renderer._finalize_interactivity()
renderer.display_inline()
Cell visualization output

Scene 4 — No Solution: Parallel Lines

When two lines have the same slope but different intercepts, they are parallel and never intersect.

$$ \begin{cases} x + y = 4 \ x + y = 2 \end{cases} $$

No common point $=$ inconsistent system.

renderer.display_all(Scene4_ParallelLines(), display=False)
renderer._finalize_interactivity()
renderer.display_inline()
Cell visualization output

Scene 5 — Infinitely Many Solutions: The Same Line Twice

When two equations are equivalent (scalar multiples of each other), they describe the same line.

$$ \begin{cases} x + y = 4 \ 2x + 2y = 8 \end{cases} $$

Every point on the line satisfies both equations — infinitely many solutions.

renderer.display_all(Scene5_InfinitelyManySolutions(), display=False)
renderer._finalize_interactivity()
renderer.display_inline()
Cell visualization output

Scene 6 — Three-Way Comparison Summary

We can classify every $2 \times 2$ linear system by looking at the geometry:

Configuration Lines Solutions
Intersecting lines Different slopes Unique solution
Parallel lines Same slope, different intercept No solution
Coincident lines Same line Infinitely many

Rule: Count common points to classify the system.

renderer.display_all(Scene6_ThreeWaySummary(), display=False)
renderer._finalize_interactivity()
renderer.display_inline()
Cell visualization output
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