Cramer’s Rule 2×2 & Diagonal-Dominant 3×3 Linear System (Ax = b) Solver Calculator

Compute system determinant D, Cramer’s determinants D_x and D_y, and exact coordinates (x, y).

Solution Coordinate x (Dx / D)
4 x (y = 2)
Solution Coordinate y (Dy / D) 2 y
System Determinant D -5 det(A)
Solution Euclidean Norm √(x² + y²) 4.472 ||x||₂
Inputs
⚡ Scenarios:
a₁
Coefficient a₁ in Equation 1 (a₁x + b₁y = c₁).
b₁
Coefficient b₁ in Equation 1.
c₁
Right-hand side c₁ in Equation 1.
a₂
Coefficient a₂ in Equation 2 (a₂x + b₂y = c₂).
b₂
Coefficient b₂ in Equation 2.
c₂
Right-hand side c₂ in Equation 2.
Results & Graph
Solution Coordinate x (Dx / D)
4 x (y = 2)
Solution Coordinate y (Dy / D)
2 y
System Determinant D
-5 det(A)
Solution Euclidean Norm √(x² + y²)
4.472 ||x||₂

Solution Coordinate x (Dx / D)

Graph 1 • Cramer’s Rule System Solver • Domain Architecture & Live Telemetry Maps Coeff A1, Coeff B1, Rhs C1 through Cramer’s Rule System Solver Formula.
Solution Coordinate x (Dx / D): 4 x (y = 2) Cramer’s Rule System Solver • Invariants & Metrics Solution Coordinate x (Dx / 4 x (y = 2) Solution Coordinate y (Dy / 2 y System Determinant D -5 det(A) Solution Euclidean Norm √(x² 4.472 ||x||₂
Graph 2 • Solution Coordinate x (Dx / D) Across Coeff A1 Sweep (40% → 160%) Evaluates D = a₁b₂ − b₁a₂; x = (c₁b₂ − b₁c₂) / D; y = (a₁c₂ − c₁a₂) / D across 7 operating points (current input highlighted).
0 x (y = 3.44 x (y = 6.88 x (y = 6.25 1.20a₁ 5.26 1.80a₁ 4.54 2.40a₁ 4 3a₁ 3.57 3.60a₁ 3.23 4.20a₁ 2.94 4.80a₁
Graph 3 • Joint Sensitivity: Coeff A1 × Coeff B1 Compares Solution Coordinate x (Dx / D) when Coeff B1 is shifted +25% (solid) vs. −25% (dashed).
0 x (y = 2.90 x (y = 5.81 x (y = --- Coeff B1 @ 75% 4.57 -30% 4.16 -15% 3.82 +0% 3.53 +15% 3.28 +30%
Summary: System determinant D = -5, Cramer numerators D_x = -20 and D_y = -10, yielding unique intersection (x, y) = (4, 2).
Summary: System determinant D = -5, Cramer numerators D_x = -20 and D_y = -10, yielding unique intersection (x, y) = (4, 2).
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Formula & Mathematical Methodology: Cramer’s Rule System Solver Formula

D = a₁b₂ − b₁a₂
x = (c₁b₂ − b₁c₂) / D
y = (a₁c₂ − c₁a₂) / D
D = a₁b₂ − b₁a₂; x = (c₁b₂ − b₁c₂) / D; y = (a₁c₂ − c₁a₂) / D
  • coeffA1: Coeff A1 (a₁)
  • coeffB1: Coeff B1 (b₁)
  • rhsC1: Rhs C1 (c₁)
  • coeffA2: Coeff A2 (a₂)
  • coeffB2: Coeff B2 (b₂)
  • rhsC2: Rhs C2 (c₂)

Rounding: Internal floating-point calculations run in 64-bit IEEE-754 double precision; display values round to 2–4 decimal places.

Worked Example: Worked Mathematical Example: Cramer’s Rule System Solver

Compute system determinant D, Cramer’s determinants D_x and D_y, and exact coordinates (x, y).

  1. Specify the function parameters, matrix entries, or confusion matrix counts.
  2. Apply the closed-form or iterative formula: D = a₁b₂ − b₁a₂; x = (c₁b₂ − b₁c₂) / D; y = (a₁c₂ − c₁a₂) / D.
  3. Examine the primary result, error bounds, and 3 interactive SVG plots.
Result: Computes Cramer’s Rule System Solver via D = a₁b₂ − b₁a₂; x = (c₁b₂ − b₁c₂) / D; y = (a₁c₂ − c₁a₂) / D.

Frequently Asked Questions

How does the Cramer’s Rule 2×2 & Diagonal-Dominant 3×3 Linear System (Ax = b) Solver Calculator compute the solution?

It evaluates D = a₁b₂ − b₁a₂; x = (c₁b₂ − b₁c₂) / D; y = (a₁c₂ − c₁a₂) / D directly in your browser and updates three custom SVG visualizations in real time.

Can I solve backwards for an input parameter using Goal Seek?

Yes. Click "🎯 Goal Seek" in the results toolbar to solve backwards for any numeric input field.

Does this calculator run locally offline?

Yes. All calculations run 100% locally in your browser and work offline via the CalculatorGraph PWA.

Mathematics & Statistics · Linear Algebra & Vectors

Cramer’s Rule 2×2 & Diagonal-Dominant 3×3 Linear System (Ax = b) Solver Calculator: inputs, method, and how to read the result

Compute system determinant D, Cramer’s determinants D_x and D_y, and exact coordinates (x, y).

Choose inputs for the case you are checking

The defaults are illustrative. Replace them with values for your case and keep the units shown beside each field.

  • Coeff A1: Coefficient a₁ in Equation 1 (a₁x + b₁y = c₁).
  • Coeff B1: Coefficient b₁ in Equation 1.
  • Rhs C1: Right-hand side c₁ in Equation 1.
  • Coeff A2: Coefficient a₂ in Equation 2 (a₂x + b₂y = c₂).

What the calculation includes

With the default values shown, Solution Coordinate x (Dx / D) is 4 x (y = 2). The calculation follows Cramer’s Rule System Solver Formula; the formula and variable definitions above show how the inputs combine.

One-input check: With the other defaults held constant, changing Coeff A1 from 3 a₁ to 4 a₁ changes Solution Coordinate x (Dx / D) from 4 x (y = 2) to 3.333 x (y = 1.333). This is a comparison inside the calculator's stated model, not a prediction.

Where to use caution

  • Uses exact analytical or standard numerical analysis formulations from NIST DLMF.
  • Double-precision floating-point arithmetic has ~15–17 significant decimal digits of precision.

Scope: Global Mathematical & Statistical Standard Check the assumptions and limitations against your situation before relying on the estimate.

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