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System of Equations Calculator

Solve a 2x2 system of linear equations using Cramer's rule.

Introduction

System of Equations Calculator — Solve a 2x2 system of linear equations using Cramer's rule. Enter a1, b1, c1, a2, b2, c2 to get an instant, accurate result.

Formula

Solves two linear equations a₁x+b₁y=c₁ and a₂x+b₂y=c₂ using Cramer's rule: x and y are found by dividing determinants of coefficient sub-matrices by the main determinant (a₁b₂−a₂b₁).

Step-by-step

  1. Enter the a1.
  2. Enter the b1.
  3. Enter the c1.
  4. Enter the a2.
  5. Enter the b2.
  6. Enter the c2.
  7. Click Calculate to see your result instantly.

Real-world example

Example: With a1 = 2, b1 = 1, c1 = 8, a2 = 1, b2 = -1, c2 = 1, the System of Equations Calculator gives X: 3, Y: 2.

Frequently Asked Questions

What if the system has no unique solution?
If the main determinant is 0, the lines are parallel or identical, and the calculator reports that no unique solution exists.
Does this handle non-linear equations?
No, this solves two linear equations in two unknowns only.
What inputs does the System of Equations Calculator need?
You'll need to provide: a1, b1, c1, a2, b2, c2. All fields use sensible defaults, so you can see a working example immediately and then adjust the values to match your own numbers.
How accurate is the System of Equations Calculator?
The System of Equations Calculator applies the formula exactly as calculated — Solves two linear equations a₁x+b₁y=c₁ and a₂x+b₂y=c₂ using Cramer's rule: x and y are found by dividing determinants of coefficient sub-matrices by the main determinant (a₁b₂−a₂b₁). — so results are precise for the inputs you provide. Accuracy depends on entering correct, realistic input values.
Is the System of Equations Calculator free to use?
Yes, the System of Equations Calculator is completely free, requires no signup, and runs instantly in your browser.

About the System of Equations Calculator

The System of Equations Calculator uses a real, verifiable formula — Solves two linear equations a₁x+b₁y=c₁ and a₂x+b₂y=c₂ using Cramer's rule: x and y are found by dividing determinants of coefficient sub-matrices by the main determinant (a₁b₂−a₂b₁). — so results are accurate every time, not an approximation.