Book Isle elimination: retain both constraints and verify equation operations

Adding two equations does not allow every other constraint to disappear. Follow an original two-variable system in Book Isle, recording operations, retained constraints and substitutions. Calculate manually or externally, without automatic solving, equivalence checking or scan-based formula recognition.

1. Fix unknown identities and both original equations

Use original material with real x and y, each letter retaining one identity across both equations. Record edition, chapter/page and phrase location, numbering equations independently of screen wrapping. Preserve coefficients, signs, equality and brackets, leaving unclear symbols unresolved. Treat the sample as abstract numbers rather than inventing incompatible physical units. Use available notes/bookmarks or verified manual transcription without equating zoom with automatic parsing.

2. Identify the unknown to eliminate

The y coefficients are one and minus one, so addition eliminates y. Add both left sides and both right sides; eliminating the left while leaving seven unchanged on the right is invalid. The consequence 3x=9 is necessary for a common solution. To preserve the full problem, replace equation one with this consequence while retaining equation two. Keeping only 3x=9 would incorrectly allow arbitrary y. Show the retained system in a two-line box.

(1) 2*x + y = 7
(2) x - y = 2
(1)+(2): 3*x = 9
Retained system: 3*x = 9 and x - y = 2

3. Solve the retained system and check both originals

Solve 3x=9 as x=3, then substitute into the retained second equation to obtain y=1. Preserve variable identity. Check originals separately: twice three plus one is seven, and three minus one is two. The candidate x=3,y=0 passes 3x=9 but fails the originals, showing why checking the consequence alone is insufficient. Keep the failed candidate and exact failed equations. Completion concerns simultaneous satisfaction of the original system, not merely correct arithmetic in the last derived line.

4. Explain why a retained equation matters

In this OpenStax elimination-method setting, subtracting the retained second equation from the sum recovers the original first equation. Record that recovery rather than an unexplained habit. Replacing both equations with duplicate sums loses information. Equivalence depends on complete corresponding operations on both sides, not arbitrary coefficient or sign changes. The reference supplies the method; readers still verify each execution.

5. Scale a whole equation for another elimination route

For another route, multiply the entire second equation by two to obtain 2x−2y=4. Subtract that from the first to get 3y=3, then retain an original equation to recover x=3. Scaling must include every term and both sides; do not transcribe minus two y as plus two y. Matching final answers cross-check this calculation but do not legitimize erroneous intermediate steps. Preserve both routes and correction history rather than covering a wrong step with another correct method.

6. Distinguish identity from contradiction

Use two separate controls. x+y=2 with 2x+2y=4 eliminates to 0=0, leaving a relation rather than a unique solution. x+y=2 with 2x+2y=6 gives 0=2, a contradiction over reals. A zero does not always mean no solution, and 0=0 does not force x=y=0. Preserve each original system and keep its result out of the initial problem. Judge all constraints, not the number of zero symbols.

7. Troubleshoot subtraction signs and copied wrapping

Subtracting original equation two from one gives x+2y=5; it does not eliminate y, because subtracting minus y changes the sign. Check operators, coefficients, equation numbers and wrapped terms. Number equations explicitly in plain text. Open exported notes and verify both originals, the retained system and both substitutions. Retrieve scans through chapter/page and phrase rather than deleting a book to fix algebra. Leave unclear symbols unresolved and distinguish personal rewrites from source quotations.

8. Rebuild retained and recovered constraints after returning

Return to the initial pair, cover results and repeat addition, retention, solving and both original checks. Explain recovery of the first equation and the difference between identity and contradiction controls. Save edition/location, domain, original system, operation/retained equation, candidates and substitutions. Another reader should detect lost conditions or signs, not rely on automatic equivalence proof. Rebuild mappings for different variables or definitions instead of transferring this answer to a similarly typeset problem.

Related tutorials and sources

Matrix row and column identity, Mathematical example notes.

Checked on November 12, 2026 against the official Book Isle App Store listing and Apple public lookup: Book Isle 1.1.4, requiring iOS / iPadOS 15.6 or later. Use existing reading, available notes or external records and verify controls in your installed version. Original samples and manual calculations are not new automatic analysis features. No physical-device, private-file or universal export-environment testing is claimed.