{"uid":"cap_xl__5BLYlu_pTQlpzjEP7","slug":"linear-algebra-system-solver-ax-b-fabb50f2","name":"Linear Algebra System Solver (Ax=b)","description":"Solve Ax=b (n up to 10) by Gaussian elimination with partial pivoting; flags singular and ill-conditioned systems.","url":"https://linear-algebra.x402tookit.com/solve?utm_source=zero.xyz","method":"GET","headers":{},"bodySchema":{"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"]}}}}}}},"responseSchema":null,"example":null,"exampleRequest":null,"tags":["x402"],"displayCostAmount":"0.02","displayCostAsset":"USDC","priceDynamic":false,"priceHint":null,"priceStatus":"priced","priceSource":"probe","requiresHandshake":false,"reviewCount":0,"rating":{"score":"0.00","successRate":"0.00","reviews":0,"stars":null,"state":"unrated"},"availabilityStatus":"unknown","priceObserved":null,"sessionDeposit":null,"pricing":{"kind":"static","summary":"$0.02/call","primary":{"kind":"static","protocol":"x402","network":"base","amountUsd":"0.02","per":"call","confidence":"exact"},"accepted":[{"kind":"static","protocol":"x402","network":"base","amountUsd":"0.02","per":"call","confidence":"exact"}]},"paymentMethods":[{"uid":"pm_MI4uVsIEIf-sPlEQAmoE3","protocol":"x402","methodType":"crypto","chain":"base","mode":"charge","costAmount":"0.02","costPer":"request","priority":0,"asset":"0x833589fCD6eDb6E08f4c7C32D4f71b54bdA02913","unit":"request","depositMicros":null,"planRef":null}],"brandName":null,"brandSlug":null,"brandBaseUrl":null,"brandDocsUrl":null,"whatItDoes":"Solves a system of linear equations Ax=b (up to 10x10) using Gaussian elimination with partial pivoting, flagging singular or ill-conditioned systems.","exampleAgentPrompt":"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].","exampleUseCases":[{"title":"Engineering circuit equation solver","prompt":"I have a circuit with three unknowns and the equations form a 3x3 system — A is [[4,-1,0],[-1,4,-1],[0,-1,4]] and b is [1,2,1]. Can you solve for the unknown voltages?"},{"title":"Detect singular system before inversion","prompt":"Before I try to invert this matrix, can you check if the system A=[[1,2,3],[4,5,6],[7,8,9]] with b=[1,2,3] is solvable or singular?"},{"title":"Solve ill-conditioned economic model","prompt":"I'm modeling a small economy with 5 equations — here's my coefficient matrix A as [[10,7,8,7,1],[7,5,6,5,1],[8,6,10,9,1],[7,5,9,10,1],[1,1,1,1,1]] and b=[32,23,33,31,5]. Can you solve it and warn me if the system might be numerically unstable?"}],"resultDescription":"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.","failureModes":["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"],"whenToPreferThis":"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.","instructions":null,"reviewSummary":null,"reviewSummaryHighlights":null,"reviewSummaryConcerns":null,"reviewSummaryGeneratedAt":null,"activationCount":0,"lastUsedAt":null,"lastSuccessfullyRanAt":null,"lastHealthCheckAt":"2026-10-30T00:34:50.199Z","isFirstParty":false,"canonicalSlug":"linear-algebra-system-solver-ax-b-fabb50f2"}