# Linear System Solver (Ax = b)

> Linear System Solver (Ax = b) is a paid API for AI agents from math-geometry.x402tookit.com, paid per call via x402, $0.02/call, status unknown (last checked 2026-11-03).

Solves a system of linear equations Ax = b for up to 6 unknowns, returning the solution vector, rank, determinant, and classification (unique/none/infinite solutions).

## Facts

- Endpoint: GET https://math-geometry.x402tookit.com/linear-system?utm_source=zero.xyz
- Price: $0.02/call
- Payment: x402
- Status: unknown
- Last checked: 2026-11-03
- Activations on Zero: 0
- Tags: x402
- Canonical page: https://www.zero.xyz/c/linear-system-solver-ax-b-e4d44427
- Structured record (JSON): https://api.zero.xyz/v1/capabilities/cap_DbwWT2JlhjgL2FjLGowtf

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 linear-system-solver-ax-b-e4d44427
```

Example prompt: Solve the linear system with coefficient matrix 2,1,-1;-3,-1,2;-2,1,2 and right-hand side values 8,-11,-3 — tell me the solution vector, the rank, and the determinant.

## When to prefer this

Use this endpoint when you need to solve a concrete numerical system of linear equations (Ax = b) with up to 6 unknowns and want both the solution and diagnostic metadata (rank, determinant, solution classification). Prefer it over a general matrix operations endpoint when the goal is solving for unknowns rather than matrix transformations. It handles the underdetermined and overdetermined cases by reporting solution type rather than failing.

## Known failure modes

- Malformed matrix string (wrong delimiter or non-numeric values) causes a parse error
- Matrix dimensions mismatch between matrix and constants returns an error
- System with more than 6 unknowns is not supported
- Singular matrix (determinant = 0) returns no-unique-solution verdict rather than a numeric answer
- Empty or missing query parameters return a 400-level error

## How this service works

Solve linear equations Ax = b (up to 6 unknowns): unique, none or infinite solutions, with rank and determinant.

## Output

Returns a JSON object containing the solution vector (x1 through xN), rank of the coefficient matrix, determinant (for square systems), number of equations and unknowns, max residual for accuracy assessment, and a summary/verdict classifying the system as having a unique solution, no solution, or infinitely many solutions.

## Request schema (JSON Schema)

```json
{
 "type": "object",
 "$schema": "https://json-schema.org/draft/2020-12/schema",
 "required": [
  "input"
 ],
 "properties": {
  "input": {
   "type": "object",
   "required": [
    "type",
    "method"
   ],
   "properties": {
    "type": {
     "type": "string",
     "const": "http"
    },
    "method": {
     "enum": [
      "GET"
     ],
     "type": "string"
    },
    "queryParams": {
     "type": "object",
     "required": [
      "matrix",
      "constants"
     ],
     "properties": {
      "matrix": {
       "type": "string",
       "description": "Coefficient matrix, rows separated by semicolons: 2,1,-1;-3,-1,2;-2,1,2"
      },
      "constants": {
       "type": "string",
       "description": "Right-hand side values: 8,-11,-3"
      }
     }
    }
   },
   "additionalProperties": false
  },
  "output": {
   "type": "object",
   "required": [
    "type"
   ],
   "properties": {
    "type": {
     "type": "string"
    },
    "example": {
     "type": "object",
     "properties": {
      "asOf": {
       "type": [
        "null",
        "string"
       ]
      },
      "source": {
       "type": [
        "null",
        "string"
       ]
      },
      "reasons": {
       "type": [
        "array",
        "null"
       ],
       "items": {
        "type": "string"
       }
      },
      "summary": {
       "type": [
        "null",
        "string"
       ]
      },
      "verdict": {
       "type": [
        "null",
        "string"
       ]
      },
      "degraded": {
       "type": [
        "boolean",
        "null"
       ]
      },
      "solution": {
       "type": [
        "null",
        "object"
       ],
       "properties": {
        "x1": {
         "type": [
          "null",
          "number"
         ]
        },
        "x2": {
         "type": [
          "null",
          "number"
         ]
        },
        "x3": {
         "type": [
          "null",
          "number"
         ]
        }
       }
      },
      "unknowns": {
       "type": [
        "null",
        "number"
       ]
      },
      "equations": {
       "type": [
        "null",
        "number"
       ]
      },
      "disclaimer": {
       "type": [
        "null",
        "string"
       ]
      },
      "notChecked": {
       "type": [
        "array",
        "null"
       ],
       "items": {
        "type": "string"
       }
      },
      "determinant": {
       "type": [
        "null",
        "number"
       ]
      },
      "maxResidual"
… (truncated)
```

## More

- Live health (JSON, refreshed every minute): https://www.zero.xyz/c/linear-system-solver-ax-b-e4d44427/health.json
- [Zero catalog index](https://www.zero.xyz/llms.txt)
- [Other services from math-geometry.x402tookit.com](https://www.zero.xyz/host/math-geometry.x402tookit.com/llms.txt)