determin.ant — Linear Algebra from Scratch

Where you'll see this: circuit analysis (Kirchhoff's laws give a system of equations), 3D rendering (ray-plane intersections), economics (supply and demand equilibrium), and every time you ask "where do these two constraints meet?" — you're solving a system of equations.

A system of linear equations is just two (or more) equations that share the same unknowns. The question is: are there values of x and y that satisfy all of them simultaneously?

Each equation is a line

Any equation of the form $ax + by = c$ describes a straight line. Every point on that line is a pair (x, y) that satisfies the equation.

So solving a system of two equations is the same as asking: where do two lines cross?

Three possible outcomes

One solution

The lines cross at exactly one point. That point is the unique pair (x, y) that satisfies both equations. Most systems look like this.

No solution

The lines are parallel — same slope, different intercepts. They never meet, so no (x, y) satisfies both at once. The system is called inconsistent.

Infinite solutions

The two equations describe the same line — one is just a multiple of the other. Every point on that line is a solution. The system is called dependent.

Try dragging the sliders to make the two lines parallel: set a₁ = a₂ and b₁ = b₂ but c₁ ≠ c₂. Watch the green dot disappear. Then set c₁ = c₂ too — the lines merge and you get infinitely many solutions.

Connection to matrices

This system can be written as a single matrix equation:

\begin{bmatrix} a_1 & b_1 \\ a_2 & b_2 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} c_1 \\ c_2 \end{bmatrix}

The matrix on the left is a transformation. Solving the system means asking: what vector, when transformed, lands on the right-hand side? If the transformation is invertible (non-zero determinant), there's exactly one answer. If not — parallel lines, det = 0 — there's either none or infinitely many.

Geometric Interpretation

Each equation is a line

Three possible outcomes

One solution

No solution

Infinite solutions

Connection to matrices