Math

System of Equations Calculator

Solve two linear equations a₁x + b₁y = c₁ and a₂x + b₂y = c₂.

a₁

b₁

c₁

a₂

b₂

c₂

Calculated Output

x 0

y 2

Determinant -5

Formula: Cramer's rule: x = Dₓ/D, y = D_y/D

Step-by-Step Breakdown

1. Values used:

2. a₁ = 2

3. b₁ = 3

4. c₁ = 6

5. a₂ = 1

6. b₂ = -1

7. c₂ = -2

9. x = 0

10. y = 2

11. Determinant = -5

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The math behind it

The determinant D = a₁b₂ − b₁a₂. If D ≠ 0 there is one solution: x = (c₁b₂ − b₁c₂)/D and y = (a₁c₂ − c₁a₂)/D. If D = 0 the lines are parallel or identical.

Worked Example

2x + 3y = 6 and x − y = −2 → x = 0, y = 2.

Frequently Asked Questions

What does a zero determinant mean?

No unique solution: the equations are either inconsistent (no solution) or dependent (infinitely many).

The math behind it

Worked Example

Frequently Asked Questions

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