Solving One Equation

This linear equation calculator solves ax + b = cx + d by collecting terms: (a − c)x = d − b, then dividing. The chart shows both sides as lines — the crossing point is the answer — and degenerate cases (0 = 0 identity, 0 = k contradiction) are detected and labeled.

Systems of Two Equations

For a₁x + b₁y = e₁ and a₂x + b₂y = e₂, Cramer's rule uses the determinant D = a₁b₂ − a₂b₁. When D ≠ 0 there is exactly one intersection; when D = 0 the lines are parallel (no solution) or coincident (infinite solutions).

Why It Matters

Linear equations model budgets, mixtures, motion, and supply-and-demand. Systems capture interactions — two constraints met simultaneously — and the geometric view on this page makes the algebra concrete.