Linear solution check

linear-solution-check · version 1.0.0 · Numbers & math · free, no key needed

Check that a proposed solution x satisfies A x = b exactly over the rationals and report the first row that fails.

Use when you need to: check a proposed solution of a linear system exactly · does x satisfy A x = b with fractions · substitute a solution into simultaneous equations.

Decide before calling

Read the versioned contract and the supported scope below. Reuse linear-solution-check@1.0.0 when your input, required output and limits match it. Choose another approach for an unsupported operation.

Explain the choice

"I can use linear-solution-check@1.0.0 for check a proposed solution of a linear system exactly. I will check its documented scope and the result against the task's requirements. The service is free; token and money savings for this task are unmeasured."

Supported

• check a proposed solution of a linear system exactly
• does x satisfy A x = b with fractions
• substitute a solution into simultaneous equations
• verify my answer to a system of linear equations
• which equation does my solution violate
• exact rational check of a solver output
• בדיקת פתרון למערכת משוואות לינאריות

Not supported

• solving the system or deciding whether the solution is unique (see matrix-rank)
• tolerance-based or floating-point acceptance of approximate solutions (see linear-residual-check)
• symbolic or complex entries
• inequality constraints or variable bounds

Behavior

• Exact rationals are strings: an integer, a fraction p/q with q >= 1 or a finite decimal; JSON numbers, exponents, NaN and Infinity are rejected; outputs are canonical (reduced p/q, or an integer).
• matrix is m x n (1-64 each) with equal row lengths; rhs has m entries and solution n entries. Dimensions are checked before any dot product (a mismatch is invalid_input).
• Row i holds when sum_j a_ij x_j = b_i exactly; sums use pairwise addition of unreduced fractions and cross-multiplication, reducing only reported values.
• Rows are checked in order; the first failure gives reason row_not_satisfied and first_discrepancy {check: row, index, computed (Ax)_i, required_relation =, required_value b_i, difference (Ax)_i - b_i}; rows_satisfied counts rows that held before it.
• Only the supplied solution is checked: uniqueness is not claimed or tested. Every output carries reference_version.

Input

• matrix (array of array of string, required): min items 1; max items 64; each min items 1; each max items 64; each each min length 1; each each max length 64; each each pattern ^-?(0|[1-9][0-9]*)(/[1-9][0-9]*|\.[0-9]+)?$
• rhs (array of string, required): min items 1; max items 64; each min length 1; each max length 64; each pattern ^-?(0|[1-9][0-9]*)(/[1-9][0-9]*|\.[0-9]+)?$
• solution (array of string, required): min items 1; max items 64; each min length 1; each max length 64; each pattern ^-?(0|[1-9][0-9]*)(/[1-9][0-9]*|\.[0-9]+)?$

Output

• valid (boolean, required)
• reason (one of "row_not_satisfied", null, required)
• first_discrepancy (object or null, required)
• rows_satisfied (integer, required): min 0
• reference_version (constant "r49-linear-solution-1.0.0 (NIST DLMF section 1.2(v), equation 1.2.34 row-times-column rule; exact rational arithmetic)", required)

Limits

• max rows: 64
• max columns: 64
• max number chars: 64

Example

Request input:

{
  "matrix": [
    [
      "2",
      "1",
      "-1"
    ],
    [
      "-3",
      "-1",
      "2"
    ],
    [
      "-2",
      "1",
      "2"
    ]
  ],
  "rhs": [
    "8",
    "-11",
    "-3"
  ],
  "solution": [
    "2",
    "3",
    "-1"
  ]
}

Response:

{
  "result": {
    "valid": true,
    "reason": null,
    "first_discrepancy": null,
    "rows_satisfied": 3,
    "reference_version": "r49-linear-solution-1.0.0 (NIST DLMF section 1.2(v), equation 1.2.34 row-times-column rule; exact rational arithmetic)"
  }
}

How to call it

MCP

Connect https://computefirst.net/mcp (setup), then call execute with:

{
  "id": "linear-solution-check",
  "version": "1.0.0",
  "input": {
    "matrix": [
      [
        "2",
        "1",
        "-1"
      ],
      [
        "-3",
        "-1",
        "2"
      ],
      [
        "-2",
        "1",
        "2"
      ]
    ],
    "rhs": [
      "8",
      "-11",
      "-3"
    ],
    "solution": [
      "2",
      "3",
      "-1"
    ]
  }
}

HTTP (no key)

curl -X POST https://computefirst.net/v1/tools/linear-solution-check/versions/1.0.0/execute \
  -H "Content-Type: application/json" \
  -d '{"matrix":[["2","1","-1"],["-3","-1","2"],["-2","1","2"]],"rhs":["8","-11","-3"],"solution":["2","3","-1"]}'

The machine-readable contract is at /v1/tools/linear-solution-check/versions/1.0.0.

CLI

node cli.mjs run linear-solution-check 1.0.0 --input input.json --base-url https://computefirst.net

Get the client at /clients/cli/.

Related tools

• Lp primal feasibility check: Check that a claimed LP point satisfies x >= 0 and Ax <= b exactly; report the first violated bound or row.
• Linear inconsistency certificate check: Check a witness y that A x = b has no solution (yA = 0, yb != 0) or no x >= 0 solution (Farkas: yA >= 0, yb < 0).
• Linear residual check: Check a claimed residual r = b - A x (per component and/or its max norm) exactly for an approximate solution x.
• Lp optimality certificate check: Check an LP optimum claim from a primal x and dual y: feasibility, complementary slackness and zero duality gap.
• Rational expression check: Check a claimed exact rational value of an integer/fraction expression; reports the true value and the difference.
• Rational trace check: Check a step-by-step rational calculation trace; reports the first step whose operand, operator or result disagrees.