# ForgeMesh Linear Equation Solver

> ForgeMesh Linear Equation Solver is a paid API for AI agents from x402.forgemesh.io, paid per call via x402, $0.002/call, status unknown (last checked 2026-09-15).

Solves linear equations, quadratic equations (real and complex roots), and 2x2 systems of linear equations given structured numeric coefficients

## Facts

- Endpoint: POST https://x402.forgemesh.io/linear-equation-solver
- Price: $0.002/call
- Payment: x402
- Status: unknown
- Last checked: 2026-09-15
- Activations on Zero: 0
- Tags: x402
- Canonical page: https://www.zero.xyz/c/forgemesh-linear-equation-solver-3fe934eb
- Structured record (JSON): https://api.zero.xyz/v1/capabilities/cap_8mWiUu99DvsWUbYVXuf2d

Status and success rate cover calls made through Zero and Zero's own probes. Third-party monitors may report differently.

## How to call it through Zero

Zero handles the 402 payment challenge and records the run. With the Zero CLI installed (`npm i -g @zeroxyz/cli`):

```sh
zero fetch --capability forgemesh-linear-equation-solver-3fe934eb -d '<json body>'
```

Example prompt: Can you solve the quadratic equation 2x² + 3x - 5 = 0 and give me the exact roots, including any complex ones if they exist?

## When to prefer this

Choose this endpoint when you need guaranteed-exact algebraic solutions to standard equation types (linear, quadratic, or 2x2 linear systems) and your coefficients are already known numerically. Prefer this over symbolic math libraries when you want a simple API call with no local computation, or over LLM-based math when you need precision and determinism. Not suitable for higher-degree polynomials, symbolic/variable coefficients, or natural language math expressions that need parsing first.

## Known failure modes

- Missing required coefficients for the chosen equation type returns a validation error
- Invalid or non-numeric coefficient strings cause parsing errors
- Unsupported equation type string returns an error
- Division by zero for degenerate cases (e.g. a=0 for quadratic) may return an error or fallback
- Network or payment failure returns a 402 or 5xx response

## How this service works

Equation solver for linear equations, quadratics with real or complex roots, and 2x2 systems of linear equations, given structured numeric coefficients rather than a text expression to parse. For engineering checks, math tutoring tools, and agents that need guaranteed-exact algebraic solutions.

## Output

Returns exact algebraic solutions: for linear equations, the value of x; for quadratics, both roots (real or complex); for 2x2 systems, the values of both variables x and y that satisfy both equations simultaneously.

## Request schema (JSON Schema)

```json
{
 "type": "object",
 "properties": {
  "a": {
   "type": "string"
  },
  "b": {
   "type": "string"
  },
  "c": {
   "type": "string"
  },
  "a1": {
   "type": "string"
  },
  "a2": {
   "type": "string"
  },
  "b1": {
   "type": "string"
  },
  "b2": {
   "type": "string"
  },
  "c1": {
   "type": "string"
  },
  "c2": {
   "type": "string"
  },
  "type": {
   "type": "string",
   "description": "linear | quadratic | system2"
  }
 }
}
```

## Response schema (JSON Schema)

```json
{
 "type": "json",
 "example": {
  "roots": [
   2,
   1
  ],
  "nature": "two real roots",
  "discriminant": 1
 }
}
```

## More

- Live health (JSON, refreshed every minute): https://www.zero.xyz/c/forgemesh-linear-equation-solver-3fe934eb/health.json
- [Zero catalog index](https://www.zero.xyz/llms.txt)
- [Other services from x402.forgemesh.io](https://www.zero.xyz/host/x402.forgemesh.io/llms.txt)