The Mathematics of Linear Systems & Cramer's Rule
A linear equation produces a straight line when graphed on a two-dimensional Cartesian plane. A system of two linear equations in two variables explores whether and where those two straight lines meet.
The Three Geometric Outcomes
| Geometric Condition | Determinant (D) | Classification | Number of Solutions |
|---|---|---|---|
| Intersecting Lines | D ≠ 0 | Consistent & Independent | Exactly One Unique Solution (x, y) |
| Parallel Lines | D = 0, (Dx ≠ 0 or Dy ≠ 0) | Inconsistent | No Solution (Lines never meet) |
| Coincident Lines (Same line) | D = 0, Dx = 0, Dy = 0 | Consistent & Dependent | Infinite Solutions (Every point on line) |
Solving via Determinants (Cramer's Rule)
D = a₁b₂ − a₂b₁
Dx = c₁b₂ − c₂b₁
Dy = a₁c₂ − a₂c₁
x = Dx / D, y = Dy / D
Frequently Asked Questions (FAQ)
What is the difference between elimination and substitution?
Substitution solves for one variable in terms of another (e.g. y = 2 - x) and substitutes it into the second equation. Elimination multiplies equations by constants so that adding or subtracting them cancels out one variable entirely.
Can decimal or negative coefficients be entered?
Yes. The solver processes positive, negative, and floating-point decimal coefficients with exact numerical precision.