Linear Systems as Intersecting Planes

System Type

Equation constants (RHS)

d₁ 0.0

d₂ 0.0

d₃ 0.0

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Solution

—

det(A)

0.00

rank(A)

Elimination steps

How it works

A 3×3 linear system Ax = b is three equations of the form a·x + b·y + c·z = d. Each equation is exactly one plane in 3D — the coefficients (a,b,c) fix its orientation (they form the plane's normal vector) and d shifts it along that normal. Solving the system geometrically means finding where all three planes meet at once:

a₁x + b₁y + c₁z = d₁   (red plane)
a₂x + b₂y + c₂z = d₂   (green plane)
a₃x + b₃y + c₃z = d₃   (blue plane)

Unique solution — the three planes meet at exactly one point, det(A) ≠ 0, rank(A) = 3. The point is found with Cramer's rule: xᵢ = det(Aᵢ)/det(A), where Aᵢ replaces column i of A with b.
No solution — two or more planes are parallel but offset (their normals are dependent, their constants are not), so no point lies on all three. det(A) = 0 and the system is inconsistent.
Infinite solutions (line) — rank(A) = 2: one equation is a combination of the others, so effectively only two independent planes remain and they intersect in a whole line, drawn in gold.
Gaussian elimination — the step log applies row operations (swap, subtract a multiple of one row from another) to reduce [A|b] to echelon form. The number of nonzero pivot rows left is the rank; a row that becomes 0 = nonzero flags an inconsistent system.

Drag the d sliders to translate a plane along its own normal without changing its tilt — watch the intersection point or line move continuously, and note that det(A) and rank(A) never change from a d-only edit, because they depend on A alone.

Plane 1 (eq. 1) Plane 2 (eq. 2) Plane 3 (eq. 3) Solution

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