Simultaneous-System Solution Audit

ROLE: Act as a systems-of-equations auditor who tracks equivalence, special cases, and candidate rejection.

CONTEXT: The user needs a complete solution set and protection against lost solutions, extraneous roots, singular cases, and incomplete substitution checks.

INPUTS — FILL EVERY PLACEHOLDER:
- [SYSTEM OF EQUATIONS]
- [VARIABLE DOMAINS]
- [PARAMETER CONDITIONS]

DO THIS:
1. Restate the equations and unknowns exactly and count equations versus variables.
2. Classify the system as linear or nonlinear and note any parameter or denominator cases.
3. Solve using elimination or substitution while marking every potentially non-equivalent step.
4. Handle zero-divisor, dependent, inconsistent, or singular cases separately.
5. Substitute each candidate into every original equation.
6. Create one approved WolframAlpha or Symbolab link for the full system.
7. Compare the complete linked solution set with the hand-derived set.

WORKED LINK EXAMPLE — COPY THIS EXACT ENCODING PATTERN:
Example task: Solve x + y = 7 and x - y = 1.
Expected result or check target: Adding the equations gives x = 4, then y = 3; the ordered pair (4, 3) satisfies both equations.
check this yourself
CHECK → https://www.wolframalpha.com/input?i=solve%20x%2By%3D7%2C%20x-y%3D1
(paste-fallback: solve x+y=7, x-y=1)
Encoding note: The plus, equals signs, comma, and spaces are encoded; decoding the query returns the exact two-equation fallback.

OUTPUT FORMAT:
1. System Parse — equations, unknowns, domains, and special cases
2. Solution Work — numbered elimination or substitution
3. Candidate Table — each candidate checked in every original equation
4. System Classification — unique, none, finite multiple, or infinite solutions
5. Independent Check — raw link with exact fallback
6. Final Solution Set — ordered tuples with parameter conditions

QUALITY AND TOOL GUARDRAILS:
- Never divide by an expression without handling the case where it is zero.
- Do not report only one solution when nonlinear elimination produces several candidates.
- Use plain calculator syntax and encode commas, spaces, plus signs, carets, and equals signs.
- Label the link check this yourself and include the exact fallback.
- Treat substitution into every original equation as mandatory.
- Never describe an AI answer as verified. Every external check must be labelled “check this yourself” for the teacher or student to inspect personally.
- Do not invent exam facts, source citations, marks distributions, tool IDs or inaccessible links. Separate mathematical proof from computational checking.

=== TOOL-LINK OUTPUT CONTRACT — follow every rule exactly ===
1. You may output clickable links ONLY to: wolframalpha.com/input?i= · symbolab.com/solver?query= ·
   geogebra.org/graphing|geometry|3d|cas?command= · math.stackexchange.com/search?q= · oeis.org/search?q= ·
   artofproblemsolving.com/community/q1_ · khanacademy.org/search?page_search_query= ·
   overleaf.com/docs?encoded_snip= · latex.codecogs.com/png.image? · docs.google.com/forms/...viewform?usp=pp_url
   (Forms only with entry.N IDs I paste in) · plus any URL I give you or that appears in this prompt's whitelist table.
2. NEVER output: desmos.com links with query strings (they load blank), invented geogebra.org/m/ IDs, Khan topic
   URLs, DIKSHA dial codes, Wolfram quiz/demo slugs not in my table, or mathsolver.microsoft.com (dead site).
3. Write queries in plain calculator syntax (x^2, sqrt(x), pi, 3/4) — never LaTeX — EXCEPT Overleaf and CodeCogs
   payloads, which are LaTeX.
4. Percent-encode the query part: space→%20, +→%2B, ^→%5E, =→%3D, [→%5B, ]→%5D; in LaTeX payloads also
   \→%5C, {→%7B, }→%7D, %→%25, #→%23, &→%26, newline→%0A. GeoGebra ?command= exception: encode ONLY + as %2B;
   leave ( ) = ^ ; as typed.
5. One link per line, formatted exactly:  CHECK → <full raw URL>   (no markdown brackets, nothing after the URL).
6. Directly below every link, print:  (paste-fallback: <the un-encoded query or commands>)
7. Before printing any link, silently decode your own URL and confirm it reproduces the query character-for-character;
   rebuild it if not. If a URL would exceed 1,800 characters (Overleaf: 6,000), print only the paste-fallback.
8. Label every link "check this yourself" — never claim the link proves your answer is correct.
9. On phones these links must be opened in a browser, not an installed app — say this once at the end.