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System of Equations Solver

Solve a 2-variable system using Cramer's rule.

Best for: Use it for two-variable problems in algebra, economics or engineering.

Complete guide

Input

How it's calculated & sources
The System of Equations Solver uses the standard, deterministic formula, so the same inputs always produce the same result. It runs entirely in your browser, so your inputs are never uploaded. Last reviewed June 2026.

Free & no sign-up · runs entirely in your browser. Results are estimates for general information, not professional advice — verify important decisions with a qualified expert. Last reviewed June 2026.

How it works

For a₁x + b₁y = c₁ and a₂x + b₂y = c₂:

x = (c₁b₂ − c₂b₁) / (a₁b₂ − a₂b₁), y = (a₁c₂ − a₂c₁) / (a₁b₂ − a₂b₁)

Example

2x + 3y = 8, x − y = 1 → x = 2.2, y = 1.2

Frequently asked questions

What if determinant is 0?+

Either no solution (parallel lines) or infinite solutions (same line).

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Complete guide

Quick answers

Short, sourced answers to the questions people (and AI assistants) ask most.

What is System of Equations Solver?
A system of equations solver finds the values that satisfy two linear equations in two unknowns at once.
Why does System of Equations Solver matter?
The point (x, y) where the two lines meet. If the denominator (a1 b2 - a2 b1) is zero, the lines are parallel (no solution) or identical (infinitely many).
How is System of Equations Solver calculated?
It uses elimination (or the equivalent determinant/Cramer's rule) to solve for x and y simultaneously. Formula: For a1x + b1y = c1 and a2x + b2y = c2: x = (c1 b2 - c2 b1) / (a1 b2 - a2 b1), and similarly for y.
What are common system of equations solver mistakes?
Missing that parallel lines have no unique solution.
When should you use the System of Equations Solver?
Use it for two-variable problems in algebra, economics or engineering.

What is the System of Equations Solver?

A system of equations solver finds the values that satisfy two linear equations in two unknowns at once.

How the System of Equations Solver works

It uses elimination (or the equivalent determinant/Cramer's rule) to solve for x and y simultaneously.

For a1x + b1y = c1 and a2x + b2y = c2: x = (c1 b2 - c2 b1) / (a1 b2 - a2 b1), and similarly for y

Common mistakes to avoid

  • Missing that parallel lines have no unique solution.
  • Arithmetic slips in the cross-multiplication.
  • Entering coefficients in the wrong positions.

Keywords: system, equation, cramer.

Reviewed by the Free Tools Galaxy editorial team · Updated June 2026 · Calculated privately in your browser.

Frequently asked questions

Is the System of Equations Solver free to use?+

Yes. Every tool on Free Tools Galaxy is 100% free, runs in your browser and requires no signup.

How accurate is the System of Equations Solver?+

System of Equations Solver uses the standard system of equations solver formula in double-precision arithmetic, so the same inputs always produce the same result and you can verify any figure by hand. It is an educational estimate — real-world outcomes depend on your actual rates, rules and assumptions.

Do you store my inputs?+

No. The System of Equations Solver runs entirely in your browser. Nothing is uploaded or saved to a server.

Can I use the System of Equations Solver on mobile?+

Yes — the interface is fully responsive and works on phones, tablets and desktops.

What are common mistakes to avoid?+

The most frequent mistake is mixing units. Double-check your inputs use a single, consistent unit before clicking Calculate.

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System of Equations Solver on Free Tools Galaxy — Solve a 2-variable system using Cramer's rule. https://freetoolsgalaxy.com/tools/system-equation-solver
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# System of Equations Solver — the free tool I keep coming back to

Solve a 2-variable system using Cramer's rule.

Try it: https://freetoolsgalaxy.com/tools/system-equation-solver
Quora answer
The fastest way is to use a free in-browser system of equations solver: https://freetoolsgalaxy.com/tools/system-equation-solver. Solve a 2-variable system using Cramer's rule.

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