Simultaneous Equations Solver (2×2) Formula

Understand the math behind the simultaneous equations solver (2×2). Each variable explained with a worked example.

Use the Simultaneous Equations Solver (2×2)

Formulas Used

Det

det = a1 * b2 - a2 * b1

X Val

x_val = (a1 * b2 - a2 * b1) != 0 ? (c1 * b2 - c2 * b1) / (a1 * b2 - a2 * b1) : 0

Y Val

y_val = (a1 * b2 - a2 * b1) != 0 ? (a1 * c2 - a2 * c1) / (a1 * b2 - a2 * b1) : 0

Variables

VariableDescriptionDefault
a1a₁ (Eq1 x-coeff)2
b1b₁ (Eq1 y-coeff)3
c1c₁ (Eq1 constant)8
a2a₂ (Eq2 x-coeff)4
b2b₂ (Eq2 y-coeff)-1
c2c₂ (Eq2 constant)2

How It Works

How to Solve Simultaneous Equations

Cramer's Rule for 2×2 Systems

Given:

  • a₁x + b₁y = c₁
  • a₂x + b₂y = c₂

    • Determinant D = a₁b₂ - a₂b₁

    x = (c₁b₂ - c₂b₁) / D

    y = (a₁c₂ - a₂c₁) / D

    If D = 0, the system has no unique solution (the lines are parallel or identical).

    Worked Example

    Solve: 2x + 3y = 8 and 4x - y = 2.

    a1 = 2b1 = 3c1 = 8a2 = 4b2 = -1c2 = 2
    1. 01D = (2)(-1) - (4)(3) = -2 - 12 = -14
    2. 02x = (8×(-1) - 2×3) / (-14) = (-8 - 6) / (-14) = -14 / -14 = 1
    3. 03y = (2×2 - 4×8) / (-14) = (4 - 32) / (-14) = -28 / -14 = 2
    4. 04Solution: x = 1, y = 2

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