Simultaneous Equations Calculator

2×2 — Linear 2×2

Elimination + Substitution + Cramer's Rule (3 full methods)

3×3 — Linear 3×3

Gaussian elimination with partial pivoting + back-substitution

f=g=0 — Nonlinear

Newton's method on 2D Jacobian — quadratic convergence

sub — Substitution

Isolate x from Eq1, substitute into Eq2, back-substitute

Ax=b — Matrix Method

A⁻¹ for 2×2; Gaussian for 3×3

r₁∩r₂ — Parametric

Grid search + Newton refinement for curve intersections

y'=Ay — ODE System

Eigenvalue/eigenvector method — node/saddle/spiral/center

∇f=λ∇g — Lagrange Mult.

3-equation Newton system; λ = shadow price

Cramer's (x) \(x=D_x/D,\;D=\det A\)

Cramer's (y) \(y=D_y/D,\;D_x=\begin{vmatrix}c_1&b_1\\c_2&b_2\end{vmatrix}\)

Elimination \(\text{Multiply rows to cancel variable}\)

2×2 Inverse \(A^{-1}=\frac{1}{ad-bc}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}\)

Newton Step \(\mathbf{x}^{k+1}=\mathbf{x}^k-J^{-1}\mathbf{F}(\mathbf{x}^k)\)

Jacobian \(J_{ij}=\partial F_i/\partial x_j\)

Row Op \(R_i\leftarrow R_i-\frac{a_{ik}}{a_{kk}}R_k\)

Char. Eq. \(\lambda^2-\mathrm{tr}(A)\lambda+\det(A)=0\)

Eigenvalue \((A-\lambda I)\mathbf{v}=\mathbf{0}\)

ODE Solution \(\mathbf{y}=C_1e^{\lambda_1 t}\mathbf{v}_1+C_2e^{\lambda_2 t}\mathbf{v}_2\)

Lagrange KKT \(\nabla f=\lambda\nabla g,\;g=k\)

Shadow Price \(\lambda=\partial f^*/\partial k\)

Parametric \(x_1(t)=x_2(s),\;y_1(t)=y_2(s)\)

Node (real λ) \(\lambda_1\cdot\lambda_2>0\Rightarrow\text{node}\)

Saddle \(\lambda_1\cdot\lambda_2<0\)

Spiral/Center \(\Delta<0\Rightarrow\text{complex}\;\lambda\)

Σ=Simultaneous Equations Calculator