# Linear Algebra System Solver (Ax=b)

> Linear Algebra System Solver (Ax=b) is a paid API for AI agents from linear-algebra.x402tookit.com, paid per call via x402, $0.02/call, status unknown (last checked 2026-10-30).

Solves a system of linear equations Ax=b (up to 10x10) using Gaussian elimination with partial pivoting, flagging singular or ill-conditioned systems.

## Facts

- Endpoint: GET https://linear-algebra.x402tookit.com/solve?utm_source=zero.xyz
- Price: $0.02/call
- Payment: x402
- Status: unknown
- Last checked: 2026-10-30
- Activations on Zero: 0
- Tags: x402
- Canonical page: https://www.zero.xyz/c/linear-algebra-system-solver-ax-b-fabb50f2
- Structured record (JSON): https://api.zero.xyz/v1/capabilities/cap_xl__5BLYlu_pTQlpzjEP7

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-algebra-system-solver-ax-b-fabb50f2
```

Example prompt: Can you solve this system of linear equations for me? The coefficient matrix A is [[2,1,-1],[-3,-1,2],[-2,1,2]] and the right-hand side vector b is [8,-11,-3].

## When to prefer this

Choose this endpoint when you need a numerical solution to a dense system of linear equations (up to 10 unknowns) and want automatic detection of singular or ill-conditioned cases. Prefer this over the matrix inverse endpoint when you only need the solution vector x (not the full inverse), or when you want to check solvability before committing to further computation. It is ideal for small to medium scientific, engineering, or educational use cases where numerical stability diagnostics (condition number, residual, digits lost) are needed alongside the solution.

## Known failure modes

- Singular system: returns singular=true with no solution vector
- Ill-conditioned system: returns illConditioned=true with high digitsLostEstimate warning
- Matrix dimension mismatch between A and b: request rejected
- Matrix size exceeds 10x10: request rejected
- Malformed JSON in A or b parameter: parse error
- Non-square matrix A: not supported

## How this service works

Solve Ax=b (n up to 10) by Gaussian elimination with partial pivoting; flags singular and ill-conditioned systems.

## Output

Returns the solution vector x (if the system is non-singular), along with a condition number (1-norm), an estimate of digits of precision lost, the maximum residual, a boolean singular flag, a boolean ill-conditioned flag, a summary verdict, and optional degraded/disclaimer notes.

## 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": [
      "A",
      "b"
     ],
     "properties": {
      "A": {
       "type": "string",
       "description": "Square coefficient matrix, JSON like [[2,1,-1],[-3,-1,2],[-2,1,2]] (max 10x10)"
      },
      "b": {
       "type": "string",
       "description": "Right-hand side vector, JSON like [8,-11,-3] or 8,-11,-3; same length as A has rows"
      }
     }
    }
   },
   "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"
       ]
      },
      "singular": {
       "type": [
        "boolean",
        "null"
       ]
      },
      "solution": {
       "type": [
        "array",
        "null"
       ],
       "items": {
        "type": "number"
       }
      },
      "disclaimer": {
       "type": [
        "null",
        "string"
       ]
      },
      "notChecked": {
       "type": [
        "array",
        "null"
       ],
       "items": {
        "type": "string"
       }
      },
      "maxResidual": {
       "type": [
        "null",
        "number"
       ]
      },
      "illConditioned": {
       "type": [
        "boolean",
        "null"
       ]
      },
      "digitsLostEstimate": {
       "type": [
        "null",
        "number"
       ]
      },
      "conditionNumber1Norm": {
       "type": [
        "null",
        "number"
       ]
      }
     }
    }
   }
  }
 }
}
```

## More

- Live health (JSON, refreshed every minute): https://www.zero.xyz/c/linear-algebra-system-solver-ax-b-fabb50f2/health.json
- [Zero catalog index](https://www.zero.xyz/llms.txt)
- [Other services from linear-algebra.x402tookit.com](https://www.zero.xyz/host/linear-algebra.x402tookit.com/llms.txt)