Linear Algebra

Systems of Equations

Two equations in two unknowns, seen two ways at once: as lines that cross, and as columns that add up to b. Elimination turns the lines about the answer without ever moving it.

x and x·a₁y and y·a₂bequation 1equation 2

Drag b, a₁ or a₂ in the column picture, or the crossing in the row picture

Solutions—

—

Coefficients A

a₁₁ 2.00

a₁₂ -1.00

a₂₁ 0.50

a₂₂ 1.50

Right-hand side b

b₁ 2.00

b₂ 2.25

Presets

Gaussian elimination

det A—

(x, y)—

rank A, rank [A | b]—

What to observe

Drag the amber tip of b. Both lines slide without turning, and the crossing's coordinates are exactly how many a₁ and a₂ it takes to reach b.
No solution: the lines are parallel, and the columns lie on one line with b off it. No mix of a₁ and a₂ ever leaves that line, and det A = 0.
Infinitely many: b sits on the columns' line (drag b near it and it snaps), the two equations become one line, and every point on it solves both. Each one is a different recipe for the same b.
Run elimination. R₂ − m·R₁ turns line 2 about the crossing until it is flat, then back-substitution turns line 1 upright: the answer never moves, while the columns end up as the axes and b as (x, y).

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