System of Linear Equations Solver (2×2)

System of Linear Equations Solver (2 x 2) is evaluated from a₁, b₁ and c₁. The calculation reports x, y and Determinant.

Results

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About the System of Linear Equations Solver (2×2)

System of Linear Equations Solver (2 x 2) is treated here as a quantitative relation between a₁, b₁, c₁ and a₂ and x, y, Determinant and System Type.

The calculator uses a multi formula configuration. Each reported value is read as a direct evaluation of the stored rules with the declared field formats and units.

Formula basis:
D = a1 x b2 - a2 x b1
x = (c1 x b2 - c2 x b1) / D
y = (a1 x c2 - a2 x c1) / D

Interpret the outputs in the order shown by the result fields. Optional inputs affect only the outputs that depend on those variables.

Formula & How It Works

The calculation applies the following relations exactly as recorded in the metadata:

D = a1 x b2 - a2 x b1
x = (c1 x b2 - c2 x b1) / D
y = (a1 x c2 - a2 x c1) / D

Each output field is produced by substituting the supplied inputs into the relevant relation and then applying the declared rounding or text format.

Worked Examples

Example 1: Supply & Demand Equilibrium

Inputs

a1: 2 b1: 1 c1: 8 a2: 1 b2: 3 c2: 13
x: 2.2. y: 3.6. Determinant: 5. System Type: Unique solution (consistent, independent)

With a₁ = 2, b₁ = 1, c₁ = 8 and a₂ = 1 as the stated inputs, the result is x = 2.2, y = 3.6 and Determinant = 5. Each value corresponds to the declared output fields.

Example 2: Mixture Problem — Alloy Percentages

Inputs

a1: 1 b1: 1 c1: 100 a2: 0.3 b2: 0.7 c2: 55
x: 37.5. y: 62.5. Determinant: 0.4. System Type: Unique solution (consistent, independent)

With a₁ = 1, b₁ = 1, c₁ = 100 and a₂ = 0.3 as the stated inputs, the result is x = 37.5, y = 62.5 and Determinant = 0.4. Each value corresponds to the declared output fields.

Example 3: Two Jobs — Hours Worked

Inputs

a1: 12 b1: 8 c1: 280 a2: 10 b2: 15 c2: 350
x: 14. y: 14. Determinant: 100. System Type: Unique solution (consistent, independent)

With a₁ = 12, b₁ = 8, c₁ = 280 and a₂ = 10 as the stated inputs, the result is x = 14, y = 14 and Determinant = 100. Each value corresponds to the declared output fields.

Example 4: No Solution — Inconsistent System

Inputs

a1: 2 b1: 4 c1: 8 a2: 1 b2: 2 c2: 5
Determinant: 0. System Type: No solution (inconsistent - parallel lines)

With a₁ = 2, b₁ = 4, c₁ = 8 and a₂ = 1 as the stated inputs, the result is Determinant = 0 and System Type = No solution (inconsistent - parallel lines). Each value corresponds to the declared output fields.

Common Use Cases

  • Solve two simultaneous equations for x and y
  • Find intersection of two lines
  • Solve mixture and rate problems algebraically
  • Solve supply and demand equilibrium