# Environments, Simultaneous Substitution, and Original Domains

This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.

This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261008-advance-20261011-r03-variable-expression-foundations · 2026-10-08 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.



## 1. Affine Expression in Two Environments

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x)=(0)x+(-3) is an expression, not an equation. Ordered x environments=[0/3, -1/2]. Give the two E values. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-3/1, -3/1)

Evaluate the same expression separately in both environments; a constant ignores x, and evaluation is not equation solving. The exact result, retaining the requested order, is (-3/1, -3/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0001/)

Variables and algebraic expressions · qf-density11-expression-two-env-values · candidate





## 2. Affine Expression in Two Environments

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x)=(-1)x+(-6) is an expression, not an equation. Ordered x environments=[1/3, -2/2]. Give the two E values. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-19/3, -5/1)

Evaluate the same expression separately in both environments; a constant ignores x, and evaluation is not equation solving. The exact result, retaining the requested order, is (-19/3, -5/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0002/)

Variables and algebraic expressions · qf-density11-expression-two-env-values · candidate





## 3. Affine Expression in Two Environments

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x)=(2)x+(-5) is an expression, not an equation. Ordered x environments=[2/3, -3/2]. Give the two E values. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-11/3, -8/1)

Evaluate the same expression separately in both environments; a constant ignores x, and evaluation is not equation solving. The exact result, retaining the requested order, is (-11/3, -8/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0003/)

Variables and algebraic expressions · qf-density11-expression-two-env-values · candidate





## 4. Affine Expression in Two Environments

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x)=(-3)x+(-4) is an expression, not an equation. Ordered x environments=[3/3, -4/2]. Give the two E values. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-7/1, 2/1)

Evaluate the same expression separately in both environments; a constant ignores x, and evaluation is not equation solving. The exact result, retaining the requested order, is (-7/1, 2/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0004/)

Variables and algebraic expressions · qf-density11-expression-two-env-values · candidate





## 5. Affine Expression in Two Environments

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x)=(4)x+(-3) is an expression, not an equation. Ordered x environments=[4/3, -5/2]. Give the two E values. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(7/3, -13/1)

Evaluate the same expression separately in both environments; a constant ignores x, and evaluation is not equation solving. The exact result, retaining the requested order, is (7/3, -13/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0005/)

Variables and algebraic expressions · qf-density11-expression-two-env-values · candidate





## 6. Affine Expression in Two Environments

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x)=(-5)x+(-2) is an expression, not an equation. Ordered x environments=[5/3, -6/2]. Give the two E values. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-31/3, 13/1)

Evaluate the same expression separately in both environments; a constant ignores x, and evaluation is not equation solving. The exact result, retaining the requested order, is (-31/3, 13/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0006/)

Variables and algebraic expressions · qf-density11-expression-two-env-values · candidate





## 7. Affine Expression in Two Environments

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x)=(6)x+(-1) is an expression, not an equation. Ordered x environments=[6/3, -7/2]. Give the two E values. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(11/1, -22/1)

Evaluate the same expression separately in both environments; a constant ignores x, and evaluation is not equation solving. The exact result, retaining the requested order, is (11/1, -22/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0007/)

Variables and algebraic expressions · qf-density11-expression-two-env-values · candidate





## 8. Affine Expression in Two Environments

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x)=(-7)x+(0) is an expression, not an equation. Ordered x environments=[7/3, -8/2]. Give the two E values. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-49/3, 28/1)

Evaluate the same expression separately in both environments; a constant ignores x, and evaluation is not equation solving. The exact result, retaining the requested order, is (-49/3, 28/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0008/)

Variables and algebraic expressions · qf-density11-expression-two-env-values · candidate





## 9. Affine Expression in Two Environments

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x)=(8)x+(1) is an expression, not an equation. Ordered x environments=[8/3, -9/2]. Give the two E values. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(67/3, -35/1)

Evaluate the same expression separately in both environments; a constant ignores x, and evaluation is not equation solving. The exact result, retaining the requested order, is (67/3, -35/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0009/)

Variables and algebraic expressions · qf-density11-expression-two-env-values · candidate





## 10. Affine Expression in Two Environments

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x)=(-9)x+(2) is an expression, not an equation. Ordered x environments=[9/3, -10/2]. Give the two E values. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-25/1, 47/1)

Evaluate the same expression separately in both environments; a constant ignores x, and evaluation is not equation solving. The exact result, retaining the requested order, is (-25/1, 47/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0010/)

Variables and algebraic expressions · qf-density11-expression-two-env-values · candidate





## 11. Affine Expression in Two Environments

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x)=(10)x+(3) is an expression, not an equation. Ordered x environments=[10/3, -11/2]. Give the two E values. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(109/3, -52/1)

Evaluate the same expression separately in both environments; a constant ignores x, and evaluation is not equation solving. The exact result, retaining the requested order, is (109/3, -52/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0011/)

Variables and algebraic expressions · qf-density11-expression-two-env-values · candidate





## 12. Affine Expression in Two Environments

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x)=(-11)x+(4) is an expression, not an equation. Ordered x environments=[11/3, -12/2]. Give the two E values. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-109/3, 70/1)

Evaluate the same expression separately in both environments; a constant ignores x, and evaluation is not equation solving. The exact result, retaining the requested order, is (-109/3, 70/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0012/)

Variables and algebraic expressions · qf-density11-expression-two-env-values · candidate





## 13. Affine Expression in Two Environments

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x)=(12)x+(5) is an expression, not an equation. Ordered x environments=[12/3, -13/2]. Give the two E values. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(53/1, -73/1)

Evaluate the same expression separately in both environments; a constant ignores x, and evaluation is not equation solving. The exact result, retaining the requested order, is (53/1, -73/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0013/)

Variables and algebraic expressions · qf-density11-expression-two-env-values · candidate





## 14. Affine Expression in Two Environments

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x)=(-13)x+(6) is an expression, not an equation. Ordered x environments=[13/3, -14/2]. Give the two E values. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-151/3, 97/1)

Evaluate the same expression separately in both environments; a constant ignores x, and evaluation is not equation solving. The exact result, retaining the requested order, is (-151/3, 97/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0014/)

Variables and algebraic expressions · qf-density11-expression-two-env-values · candidate





## 15. Affine Expression in Two Environments

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x)=(14)x+(7) is an expression, not an equation. Ordered x environments=[14/3, -15/2]. Give the two E values. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(217/3, -98/1)

Evaluate the same expression separately in both environments; a constant ignores x, and evaluation is not equation solving. The exact result, retaining the requested order, is (217/3, -98/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0015/)

Variables and algebraic expressions · qf-density11-expression-two-env-values · candidate





## 16. Affine Expression in Two Environments

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x)=(-15)x+(8) is an expression, not an equation. Ordered x environments=[15/3, -16/2]. Give the two E values. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-67/1, 128/1)

Evaluate the same expression separately in both environments; a constant ignores x, and evaluation is not equation solving. The exact result, retaining the requested order, is (-67/1, 128/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0016/)

Variables and algebraic expressions · qf-density11-expression-two-env-values · candidate





## 17. Affine Expression in Two Environments

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x)=(16)x+(9) is an expression, not an equation. Ordered x environments=[16/3, -17/2]. Give the two E values. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(283/3, -127/1)

Evaluate the same expression separately in both environments; a constant ignores x, and evaluation is not equation solving. The exact result, retaining the requested order, is (283/3, -127/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0017/)

Variables and algebraic expressions · qf-density11-expression-two-env-values · candidate





## 18. Affine Expression in Two Environments

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x)=(-17)x+(10) is an expression, not an equation. Ordered x environments=[17/3, -18/2]. Give the two E values. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-259/3, 163/1)

Evaluate the same expression separately in both environments; a constant ignores x, and evaluation is not equation solving. The exact result, retaining the requested order, is (-259/3, 163/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0018/)

Variables and algebraic expressions · qf-density11-expression-two-env-values · candidate





## 19. Affine Expression in Two Environments

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x)=(18)x+(11) is an expression, not an equation. Ordered x environments=[18/3, -19/2]. Give the two E values. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(119/1, -160/1)

Evaluate the same expression separately in both environments; a constant ignores x, and evaluation is not equation solving. The exact result, retaining the requested order, is (119/1, -160/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0019/)

Variables and algebraic expressions · qf-density11-expression-two-env-values · candidate





## 20. Affine Expression in Two Environments

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x)=(-19)x+(12) is an expression, not an equation. Ordered x environments=[19/3, -20/2]. Give the two E values. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-325/3, 202/1)

Evaluate the same expression separately in both environments; a constant ignores x, and evaluation is not equation solving. The exact result, retaining the requested order, is (-325/3, 202/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0020/)

Variables and algebraic expressions · qf-density11-expression-two-env-values · candidate





## 21. Affine Expression in Two Environments

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x)=(20)x+(13) is an expression, not an equation. Ordered x environments=[20/3, -21/2]. Give the two E values. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(439/3, -197/1)

Evaluate the same expression separately in both environments; a constant ignores x, and evaluation is not equation solving. The exact result, retaining the requested order, is (439/3, -197/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0021/)

Variables and algebraic expressions · qf-density11-expression-two-env-values · candidate





## 22. Affine Expression in Two Environments

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x)=(-21)x+(14) is an expression, not an equation. Ordered x environments=[21/3, -22/2]. Give the two E values. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-133/1, 245/1)

Evaluate the same expression separately in both environments; a constant ignores x, and evaluation is not equation solving. The exact result, retaining the requested order, is (-133/1, 245/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0022/)

Variables and algebraic expressions · qf-density11-expression-two-env-values · candidate





## 23. Affine Expression in Two Environments

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x)=(22)x+(15) is an expression, not an equation. Ordered x environments=[22/3, -23/2]. Give the two E values. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(529/3, -238/1)

Evaluate the same expression separately in both environments; a constant ignores x, and evaluation is not equation solving. The exact result, retaining the requested order, is (529/3, -238/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0023/)

Variables and algebraic expressions · qf-density11-expression-two-env-values · candidate





## 24. Affine Expression in Two Environments

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x)=(-23)x+(16) is an expression, not an equation. Ordered x environments=[23/3, -24/2]. Give the two E values. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-481/3, 292/1)

Evaluate the same expression separately in both environments; a constant ignores x, and evaluation is not equation solving. The exact result, retaining the requested order, is (-481/3, 292/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0024/)

Variables and algebraic expressions · qf-density11-expression-two-env-values · candidate





## 25. Affine Expression in Two Environments

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x)=(24)x+(17) is an expression, not an equation. Ordered x environments=[24/3, -25/2]. Give the two E values. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(209/1, -283/1)

Evaluate the same expression separately in both environments; a constant ignores x, and evaluation is not equation solving. The exact result, retaining the requested order, is (209/1, -283/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0025/)

Variables and algebraic expressions · qf-density11-expression-two-env-values · candidate





## 26. Simultaneous Variable Swap

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x,y)=(0)x+(0)y, x=1, y=2. Swap x and y simultaneously; give (original value, swapped value) as integers.

View answer and explanation(0, 0)

Both new bindings come from the original environment; sequential overwriting is not simultaneous substitution. The exact result, retaining the requested order, is (0, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0026/)

Variables and algebraic expressions · qf-density11-expression-simultaneous-swap · candidate





## 27. Simultaneous Variable Swap

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x,y)=(2)x+(2)y, x=-1, y=3. Swap x and y simultaneously; give (original value, swapped value) as integers.

View answer and explanation(4, 4)

Both new bindings come from the original environment; sequential overwriting is not simultaneous substitution. The exact result, retaining the requested order, is (4, 4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0027/)

Variables and algebraic expressions · qf-density11-expression-simultaneous-swap · candidate





## 28. Simultaneous Variable Swap

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x,y)=(1)x+(-2)y, x=3, y=3. Swap x and y simultaneously; give (original value, swapped value) as integers.

View answer and explanation(-3, -3)

Both new bindings come from the original environment; sequential overwriting is not simultaneous substitution. The exact result, retaining the requested order, is (-3, -3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0028/)

Variables and algebraic expressions · qf-density11-expression-simultaneous-swap · candidate





## 29. Simultaneous Variable Swap

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x,y)=(4)x+(-3)y, x=1, y=6. Swap x and y simultaneously; give (original value, swapped value) as integers.

View answer and explanation(-14, 21)

Both new bindings come from the original environment; sequential overwriting is not simultaneous substitution. The exact result, retaining the requested order, is (-14, 21).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0029/)

Variables and algebraic expressions · qf-density11-expression-simultaneous-swap · candidate





## 30. Simultaneous Variable Swap

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x,y)=(5)x+(-4)y, x=2, y=7. Swap x and y simultaneously; give (original value, swapped value) as integers.

View answer and explanation(-18, 27)

Both new bindings come from the original environment; sequential overwriting is not simultaneous substitution. The exact result, retaining the requested order, is (-18, 27).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0030/)

Variables and algebraic expressions · qf-density11-expression-simultaneous-swap · candidate





## 31. Simultaneous Variable Swap

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x,y)=(6)x+(-5)y, x=-2, y=8. Swap x and y simultaneously; give (original value, swapped value) as integers.

View answer and explanation(-52, 58)

Both new bindings come from the original environment; sequential overwriting is not simultaneous substitution. The exact result, retaining the requested order, is (-52, 58).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0031/)

Variables and algebraic expressions · qf-density11-expression-simultaneous-swap · candidate





## 32. Simultaneous Variable Swap

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x,y)=(7)x+(-6)y, x=-1, y=9. Swap x and y simultaneously; give (original value, swapped value) as integers.

View answer and explanation(-61, 69)

Both new bindings come from the original environment; sequential overwriting is not simultaneous substitution. The exact result, retaining the requested order, is (-61, 69).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0032/)

Variables and algebraic expressions · qf-density11-expression-simultaneous-swap · candidate





## 33. Simultaneous Variable Swap

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x,y)=(8)x+(-7)y, x=0, y=10. Swap x and y simultaneously; give (original value, swapped value) as integers.

View answer and explanation(-70, 80)

Both new bindings come from the original environment; sequential overwriting is not simultaneous substitution. The exact result, retaining the requested order, is (-70, 80).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0033/)

Variables and algebraic expressions · qf-density11-expression-simultaneous-swap · candidate





## 34. Simultaneous Variable Swap

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x,y)=(9)x+(-8)y, x=1, y=11. Swap x and y simultaneously; give (original value, swapped value) as integers.

View answer and explanation(-79, 91)

Both new bindings come from the original environment; sequential overwriting is not simultaneous substitution. The exact result, retaining the requested order, is (-79, 91).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0034/)

Variables and algebraic expressions · qf-density11-expression-simultaneous-swap · candidate





## 35. Simultaneous Variable Swap

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x,y)=(10)x+(-9)y, x=2, y=12. Swap x and y simultaneously; give (original value, swapped value) as integers.

View answer and explanation(-88, 102)

Both new bindings come from the original environment; sequential overwriting is not simultaneous substitution. The exact result, retaining the requested order, is (-88, 102).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0035/)

Variables and algebraic expressions · qf-density11-expression-simultaneous-swap · candidate





## 36. Simultaneous Variable Swap

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x,y)=(11)x+(-10)y, x=-2, y=13. Swap x and y simultaneously; give (original value, swapped value) as integers.

View answer and explanation(-152, 163)

Both new bindings come from the original environment; sequential overwriting is not simultaneous substitution. The exact result, retaining the requested order, is (-152, 163).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0036/)

Variables and algebraic expressions · qf-density11-expression-simultaneous-swap · candidate





## 37. Simultaneous Variable Swap

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x,y)=(12)x+(-11)y, x=-1, y=14. Swap x and y simultaneously; give (original value, swapped value) as integers.

View answer and explanation(-166, 179)

Both new bindings come from the original environment; sequential overwriting is not simultaneous substitution. The exact result, retaining the requested order, is (-166, 179).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0037/)

Variables and algebraic expressions · qf-density11-expression-simultaneous-swap · candidate





## 38. Simultaneous Variable Swap

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x,y)=(13)x+(-12)y, x=0, y=15. Swap x and y simultaneously; give (original value, swapped value) as integers.

View answer and explanation(-180, 195)

Both new bindings come from the original environment; sequential overwriting is not simultaneous substitution. The exact result, retaining the requested order, is (-180, 195).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0038/)

Variables and algebraic expressions · qf-density11-expression-simultaneous-swap · candidate





## 39. Simultaneous Variable Swap

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x,y)=(14)x+(-13)y, x=1, y=16. Swap x and y simultaneously; give (original value, swapped value) as integers.

View answer and explanation(-194, 211)

Both new bindings come from the original environment; sequential overwriting is not simultaneous substitution. The exact result, retaining the requested order, is (-194, 211).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0039/)

Variables and algebraic expressions · qf-density11-expression-simultaneous-swap · candidate





## 40. Simultaneous Variable Swap

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x,y)=(15)x+(-14)y, x=2, y=17. Swap x and y simultaneously; give (original value, swapped value) as integers.

View answer and explanation(-208, 227)

Both new bindings come from the original environment; sequential overwriting is not simultaneous substitution. The exact result, retaining the requested order, is (-208, 227).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0040/)

Variables and algebraic expressions · qf-density11-expression-simultaneous-swap · candidate





## 41. Simultaneous Variable Swap

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x,y)=(16)x+(-15)y, x=-2, y=18. Swap x and y simultaneously; give (original value, swapped value) as integers.

View answer and explanation(-302, 318)

Both new bindings come from the original environment; sequential overwriting is not simultaneous substitution. The exact result, retaining the requested order, is (-302, 318).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0041/)

Variables and algebraic expressions · qf-density11-expression-simultaneous-swap · candidate





## 42. Simultaneous Variable Swap

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x,y)=(17)x+(-16)y, x=-1, y=19. Swap x and y simultaneously; give (original value, swapped value) as integers.

View answer and explanation(-321, 339)

Both new bindings come from the original environment; sequential overwriting is not simultaneous substitution. The exact result, retaining the requested order, is (-321, 339).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0042/)

Variables and algebraic expressions · qf-density11-expression-simultaneous-swap · candidate





## 43. Simultaneous Variable Swap

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x,y)=(18)x+(-17)y, x=0, y=20. Swap x and y simultaneously; give (original value, swapped value) as integers.

View answer and explanation(-340, 360)

Both new bindings come from the original environment; sequential overwriting is not simultaneous substitution. The exact result, retaining the requested order, is (-340, 360).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0043/)

Variables and algebraic expressions · qf-density11-expression-simultaneous-swap · candidate





## 44. Simultaneous Variable Swap

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x,y)=(19)x+(-18)y, x=1, y=21. Swap x and y simultaneously; give (original value, swapped value) as integers.

View answer and explanation(-359, 381)

Both new bindings come from the original environment; sequential overwriting is not simultaneous substitution. The exact result, retaining the requested order, is (-359, 381).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0044/)

Variables and algebraic expressions · qf-density11-expression-simultaneous-swap · candidate





## 45. Simultaneous Variable Swap

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x,y)=(20)x+(-19)y, x=2, y=22. Swap x and y simultaneously; give (original value, swapped value) as integers.

View answer and explanation(-378, 402)

Both new bindings come from the original environment; sequential overwriting is not simultaneous substitution. The exact result, retaining the requested order, is (-378, 402).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0045/)

Variables and algebraic expressions · qf-density11-expression-simultaneous-swap · candidate





## 46. Simultaneous Variable Swap

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x,y)=(21)x+(-20)y, x=-2, y=23. Swap x and y simultaneously; give (original value, swapped value) as integers.

View answer and explanation(-502, 523)

Both new bindings come from the original environment; sequential overwriting is not simultaneous substitution. The exact result, retaining the requested order, is (-502, 523).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0046/)

Variables and algebraic expressions · qf-density11-expression-simultaneous-swap · candidate





## 47. Simultaneous Variable Swap

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x,y)=(22)x+(-21)y, x=-1, y=24. Swap x and y simultaneously; give (original value, swapped value) as integers.

View answer and explanation(-526, 549)

Both new bindings come from the original environment; sequential overwriting is not simultaneous substitution. The exact result, retaining the requested order, is (-526, 549).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0047/)

Variables and algebraic expressions · qf-density11-expression-simultaneous-swap · candidate





## 48. Simultaneous Variable Swap

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x,y)=(23)x+(-22)y, x=0, y=25. Swap x and y simultaneously; give (original value, swapped value) as integers.

View answer and explanation(-550, 575)

Both new bindings come from the original environment; sequential overwriting is not simultaneous substitution. The exact result, retaining the requested order, is (-550, 575).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0048/)

Variables and algebraic expressions · qf-density11-expression-simultaneous-swap · candidate





## 49. Simultaneous Variable Swap

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x,y)=(24)x+(-23)y, x=1, y=26. Swap x and y simultaneously; give (original value, swapped value) as integers.

View answer and explanation(-574, 601)

Both new bindings come from the original environment; sequential overwriting is not simultaneous substitution. The exact result, retaining the requested order, is (-574, 601).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0049/)

Variables and algebraic expressions · qf-density11-expression-simultaneous-swap · candidate





## 50. Simultaneous Variable Swap

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x,y)=(25)x+(-24)y, x=2, y=27. Swap x and y simultaneously; give (original value, swapped value) as integers.

View answer and explanation(-598, 627)

Both new bindings come from the original environment; sequential overwriting is not simultaneous substitution. The exact result, retaining the requested order, is (-598, 627).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0050/)

Variables and algebraic expressions · qf-density11-expression-simultaneous-swap · candidate





## 51. Outer and Lexically Shadowed Values

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. Outer x=0, locally shadowing inner x=0, c=1. Give (outer x+c, inner x+c) as integers; the inner binding does not mutate the outer one.

View answer and explanation(1, 1)

Use each scope’s own binding before adding c; lexical shadowing is not assignment to the outer environment. The exact result, retaining the requested order, is (1, 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0051/)

Variables and algebraic expressions · qf-density11-expression-local-shadow-pair · candidate





## 52. Outer and Lexically Shadowed Values

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. Outer x=3, locally shadowing inner x=-2, c=4. Give (outer x+c, inner x+c) as integers; the inner binding does not mutate the outer one.

View answer and explanation(7, 2)

Use each scope’s own binding before adding c; lexical shadowing is not assignment to the outer environment. The exact result, retaining the requested order, is (7, 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0052/)

Variables and algebraic expressions · qf-density11-expression-local-shadow-pair · candidate





## 53. Outer and Lexically Shadowed Values

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. Outer x=-10, locally shadowing inner x=-1, c=0. Give (outer x+c, inner x+c) as integers; the inner binding does not mutate the outer one.

View answer and explanation(-10, -1)

Use each scope’s own binding before adding c; lexical shadowing is not assignment to the outer environment. The exact result, retaining the requested order, is (-10, -1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0053/)

Variables and algebraic expressions · qf-density11-expression-local-shadow-pair · candidate





## 54. Outer and Lexically Shadowed Values

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. Outer x=-9, locally shadowing inner x=-2, c=1. Give (outer x+c, inner x+c) as integers; the inner binding does not mutate the outer one.

View answer and explanation(-8, -1)

Use each scope’s own binding before adding c; lexical shadowing is not assignment to the outer environment. The exact result, retaining the requested order, is (-8, -1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0054/)

Variables and algebraic expressions · qf-density11-expression-local-shadow-pair · candidate





## 55. Outer and Lexically Shadowed Values

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. Outer x=-8, locally shadowing inner x=-3, c=-2. Give (outer x+c, inner x+c) as integers; the inner binding does not mutate the outer one.

View answer and explanation(-10, -5)

Use each scope’s own binding before adding c; lexical shadowing is not assignment to the outer environment. The exact result, retaining the requested order, is (-10, -5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0055/)

Variables and algebraic expressions · qf-density11-expression-local-shadow-pair · candidate





## 56. Outer and Lexically Shadowed Values

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. Outer x=-7, locally shadowing inner x=-4, c=-1. Give (outer x+c, inner x+c) as integers; the inner binding does not mutate the outer one.

View answer and explanation(-8, -5)

Use each scope’s own binding before adding c; lexical shadowing is not assignment to the outer environment. The exact result, retaining the requested order, is (-8, -5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0056/)

Variables and algebraic expressions · qf-density11-expression-local-shadow-pair · candidate





## 57. Outer and Lexically Shadowed Values

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. Outer x=-6, locally shadowing inner x=-5, c=0. Give (outer x+c, inner x+c) as integers; the inner binding does not mutate the outer one.

View answer and explanation(-6, -5)

Use each scope’s own binding before adding c; lexical shadowing is not assignment to the outer environment. The exact result, retaining the requested order, is (-6, -5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0057/)

Variables and algebraic expressions · qf-density11-expression-local-shadow-pair · candidate





## 58. Outer and Lexically Shadowed Values

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. Outer x=-5, locally shadowing inner x=-6, c=1. Give (outer x+c, inner x+c) as integers; the inner binding does not mutate the outer one.

View answer and explanation(-4, -5)

Use each scope’s own binding before adding c; lexical shadowing is not assignment to the outer environment. The exact result, retaining the requested order, is (-4, -5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0058/)

Variables and algebraic expressions · qf-density11-expression-local-shadow-pair · candidate





## 59. Outer and Lexically Shadowed Values

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. Outer x=-4, locally shadowing inner x=-7, c=-2. Give (outer x+c, inner x+c) as integers; the inner binding does not mutate the outer one.

View answer and explanation(-6, -9)

Use each scope’s own binding before adding c; lexical shadowing is not assignment to the outer environment. The exact result, retaining the requested order, is (-6, -9).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0059/)

Variables and algebraic expressions · qf-density11-expression-local-shadow-pair · candidate





## 60. Outer and Lexically Shadowed Values

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. Outer x=-3, locally shadowing inner x=-8, c=-1. Give (outer x+c, inner x+c) as integers; the inner binding does not mutate the outer one.

View answer and explanation(-4, -9)

Use each scope’s own binding before adding c; lexical shadowing is not assignment to the outer environment. The exact result, retaining the requested order, is (-4, -9).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0060/)

Variables and algebraic expressions · qf-density11-expression-local-shadow-pair · candidate





## 61. Outer and Lexically Shadowed Values

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. Outer x=-2, locally shadowing inner x=-9, c=0. Give (outer x+c, inner x+c) as integers; the inner binding does not mutate the outer one.

View answer and explanation(-2, -9)

Use each scope’s own binding before adding c; lexical shadowing is not assignment to the outer environment. The exact result, retaining the requested order, is (-2, -9).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0061/)

Variables and algebraic expressions · qf-density11-expression-local-shadow-pair · candidate





## 62. Outer and Lexically Shadowed Values

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. Outer x=-1, locally shadowing inner x=-10, c=1. Give (outer x+c, inner x+c) as integers; the inner binding does not mutate the outer one.

View answer and explanation(0, -9)

Use each scope’s own binding before adding c; lexical shadowing is not assignment to the outer environment. The exact result, retaining the requested order, is (0, -9).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0062/)

Variables and algebraic expressions · qf-density11-expression-local-shadow-pair · candidate





## 63. Outer and Lexically Shadowed Values

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. Outer x=0, locally shadowing inner x=-11, c=-2. Give (outer x+c, inner x+c) as integers; the inner binding does not mutate the outer one.

View answer and explanation(-2, -13)

Use each scope’s own binding before adding c; lexical shadowing is not assignment to the outer environment. The exact result, retaining the requested order, is (-2, -13).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0063/)

Variables and algebraic expressions · qf-density11-expression-local-shadow-pair · candidate





## 64. Outer and Lexically Shadowed Values

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. Outer x=1, locally shadowing inner x=-12, c=-1. Give (outer x+c, inner x+c) as integers; the inner binding does not mutate the outer one.

View answer and explanation(0, -13)

Use each scope’s own binding before adding c; lexical shadowing is not assignment to the outer environment. The exact result, retaining the requested order, is (0, -13).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0064/)

Variables and algebraic expressions · qf-density11-expression-local-shadow-pair · candidate





## 65. Outer and Lexically Shadowed Values

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. Outer x=2, locally shadowing inner x=-13, c=0. Give (outer x+c, inner x+c) as integers; the inner binding does not mutate the outer one.

View answer and explanation(2, -13)

Use each scope’s own binding before adding c; lexical shadowing is not assignment to the outer environment. The exact result, retaining the requested order, is (2, -13).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0065/)

Variables and algebraic expressions · qf-density11-expression-local-shadow-pair · candidate





## 66. Outer and Lexically Shadowed Values

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. Outer x=3, locally shadowing inner x=-14, c=1. Give (outer x+c, inner x+c) as integers; the inner binding does not mutate the outer one.

View answer and explanation(4, -13)

Use each scope’s own binding before adding c; lexical shadowing is not assignment to the outer environment. The exact result, retaining the requested order, is (4, -13).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0066/)

Variables and algebraic expressions · qf-density11-expression-local-shadow-pair · candidate





## 67. Outer and Lexically Shadowed Values

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. Outer x=4, locally shadowing inner x=-15, c=-2. Give (outer x+c, inner x+c) as integers; the inner binding does not mutate the outer one.

View answer and explanation(2, -17)

Use each scope’s own binding before adding c; lexical shadowing is not assignment to the outer environment. The exact result, retaining the requested order, is (2, -17).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0067/)

Variables and algebraic expressions · qf-density11-expression-local-shadow-pair · candidate





## 68. Outer and Lexically Shadowed Values

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. Outer x=5, locally shadowing inner x=-16, c=-1. Give (outer x+c, inner x+c) as integers; the inner binding does not mutate the outer one.

View answer and explanation(4, -17)

Use each scope’s own binding before adding c; lexical shadowing is not assignment to the outer environment. The exact result, retaining the requested order, is (4, -17).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0068/)

Variables and algebraic expressions · qf-density11-expression-local-shadow-pair · candidate





## 69. Outer and Lexically Shadowed Values

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. Outer x=6, locally shadowing inner x=-17, c=0. Give (outer x+c, inner x+c) as integers; the inner binding does not mutate the outer one.

View answer and explanation(6, -17)

Use each scope’s own binding before adding c; lexical shadowing is not assignment to the outer environment. The exact result, retaining the requested order, is (6, -17).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0069/)

Variables and algebraic expressions · qf-density11-expression-local-shadow-pair · candidate





## 70. Outer and Lexically Shadowed Values

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. Outer x=7, locally shadowing inner x=-18, c=1. Give (outer x+c, inner x+c) as integers; the inner binding does not mutate the outer one.

View answer and explanation(8, -17)

Use each scope’s own binding before adding c; lexical shadowing is not assignment to the outer environment. The exact result, retaining the requested order, is (8, -17).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0070/)

Variables and algebraic expressions · qf-density11-expression-local-shadow-pair · candidate





## 71. Outer and Lexically Shadowed Values

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. Outer x=8, locally shadowing inner x=-19, c=-2. Give (outer x+c, inner x+c) as integers; the inner binding does not mutate the outer one.

View answer and explanation(6, -21)

Use each scope’s own binding before adding c; lexical shadowing is not assignment to the outer environment. The exact result, retaining the requested order, is (6, -21).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0071/)

Variables and algebraic expressions · qf-density11-expression-local-shadow-pair · candidate





## 72. Outer and Lexically Shadowed Values

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. Outer x=9, locally shadowing inner x=-20, c=-1. Give (outer x+c, inner x+c) as integers; the inner binding does not mutate the outer one.

View answer and explanation(8, -21)

Use each scope’s own binding before adding c; lexical shadowing is not assignment to the outer environment. The exact result, retaining the requested order, is (8, -21).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0072/)

Variables and algebraic expressions · qf-density11-expression-local-shadow-pair · candidate





## 73. Outer and Lexically Shadowed Values

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. Outer x=10, locally shadowing inner x=-21, c=0. Give (outer x+c, inner x+c) as integers; the inner binding does not mutate the outer one.

View answer and explanation(10, -21)

Use each scope’s own binding before adding c; lexical shadowing is not assignment to the outer environment. The exact result, retaining the requested order, is (10, -21).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0073/)

Variables and algebraic expressions · qf-density11-expression-local-shadow-pair · candidate





## 74. Outer and Lexically Shadowed Values

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. Outer x=11, locally shadowing inner x=-22, c=1. Give (outer x+c, inner x+c) as integers; the inner binding does not mutate the outer one.

View answer and explanation(12, -21)

Use each scope’s own binding before adding c; lexical shadowing is not assignment to the outer environment. The exact result, retaining the requested order, is (12, -21).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0074/)

Variables and algebraic expressions · qf-density11-expression-local-shadow-pair · candidate





## 75. Outer and Lexically Shadowed Values

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. Outer x=12, locally shadowing inner x=-23, c=-2. Give (outer x+c, inner x+c) as integers; the inner binding does not mutate the outer one.

View answer and explanation(10, -25)

Use each scope’s own binding before adding c; lexical shadowing is not assignment to the outer environment. The exact result, retaining the requested order, is (10, -25).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0075/)

Variables and algebraic expressions · qf-density11-expression-local-shadow-pair · candidate





## 76. Original Rational-Expression Domain Guard

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. The original expression is E(x)=(0)/(x−(1)), excluding x=1. Ordered candidates=[1/1, 2/1, 0/1]. Give (three domain-eligibility values, three E values or undefined).

View answer and explanation([No, Yes, Yes], [undefined, 0/1, 0/1])

Check the original excluded point before evaluating; even a zero numerator cannot fill a denominator-zero hole or extend the original domain. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 0/1, 0/1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0076/)

Variables and algebraic expressions · qf-density11-expression-domain-candidate · candidate





## 77. Original Rational-Expression Domain Guard

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. The original expression is E(x)=(5)/(x−(-2)), excluding x=-2. Ordered candidates=[-2/1, -2/1, -2/1]. Give (three domain-eligibility values, three E values or undefined).

View answer and explanation([No, No, No], [undefined, undefined, undefined])

Check the original excluded point before evaluating; even a zero numerator cannot fill a denominator-zero hole or extend the original domain. The exact result, retaining the requested order, is ([No, No, No], [undefined, undefined, undefined]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0077/)

Variables and algebraic expressions · qf-density11-expression-domain-candidate · candidate





## 78. Original Rational-Expression Domain Guard

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. The original expression is E(x)=(2)/(x−(0)), excluding x=0. Ordered candidates=[0/1, 1/1, -1/1]. Give (three domain-eligibility values, three E values or undefined).

View answer and explanation([No, Yes, Yes], [undefined, 2/1, -2/1])

Check the original excluded point before evaluating; even a zero numerator cannot fill a denominator-zero hole or extend the original domain. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 2/1, -2/1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0078/)

Variables and algebraic expressions · qf-density11-expression-domain-candidate · candidate





## 79. Original Rational-Expression Domain Guard

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. The original expression is E(x)=(2)/(x−(1)), excluding x=1. Ordered candidates=[1/1, 6/2, -4/1]. Give (three domain-eligibility values, three E values or undefined).

View answer and explanation([No, Yes, Yes], [undefined, 1/1, -2/5])

Check the original excluded point before evaluating; even a zero numerator cannot fill a denominator-zero hole or extend the original domain. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 1/1, -2/5]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0079/)

Variables and algebraic expressions · qf-density11-expression-domain-candidate · candidate





## 80. Original Rational-Expression Domain Guard

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. The original expression is E(x)=(3)/(x−(2)), excluding x=2. Ordered candidates=[2/1, 9/2, -4/1]. Give (three domain-eligibility values, three E values or undefined).

View answer and explanation([No, Yes, Yes], [undefined, 6/5, -1/2])

Check the original excluded point before evaluating; even a zero numerator cannot fill a denominator-zero hole or extend the original domain. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 6/5, -1/2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0080/)

Variables and algebraic expressions · qf-density11-expression-domain-candidate · candidate





## 81. Original Rational-Expression Domain Guard

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. The original expression is E(x)=(4)/(x−(-2)), excluding x=-2. Ordered candidates=[-2/1, 2/2, -9/1]. Give (three domain-eligibility values, three E values or undefined).

View answer and explanation([No, Yes, Yes], [undefined, 4/3, -4/7])

Check the original excluded point before evaluating; even a zero numerator cannot fill a denominator-zero hole or extend the original domain. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 4/3, -4/7]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0081/)

Variables and algebraic expressions · qf-density11-expression-domain-candidate · candidate





## 82. Original Rational-Expression Domain Guard

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. The original expression is E(x)=(5)/(x−(-1)), excluding x=-1. Ordered candidates=[-1/1, 5/2, -9/1]. Give (three domain-eligibility values, three E values or undefined).

View answer and explanation([No, Yes, Yes], [undefined, 10/7, -5/8])

Check the original excluded point before evaluating; even a zero numerator cannot fill a denominator-zero hole or extend the original domain. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 10/7, -5/8]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0082/)

Variables and algebraic expressions · qf-density11-expression-domain-candidate · candidate





## 83. Original Rational-Expression Domain Guard

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. The original expression is E(x)=(6)/(x−(0)), excluding x=0. Ordered candidates=[0/1, 8/2, -9/1]. Give (three domain-eligibility values, three E values or undefined).

View answer and explanation([No, Yes, Yes], [undefined, 3/2, -2/3])

Check the original excluded point before evaluating; even a zero numerator cannot fill a denominator-zero hole or extend the original domain. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 3/2, -2/3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0083/)

Variables and algebraic expressions · qf-density11-expression-domain-candidate · candidate





## 84. Original Rational-Expression Domain Guard

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. The original expression is E(x)=(7)/(x−(1)), excluding x=1. Ordered candidates=[1/1, 11/2, -9/1]. Give (three domain-eligibility values, three E values or undefined).

View answer and explanation([No, Yes, Yes], [undefined, 14/9, -7/10])

Check the original excluded point before evaluating; even a zero numerator cannot fill a denominator-zero hole or extend the original domain. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 14/9, -7/10]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0084/)

Variables and algebraic expressions · qf-density11-expression-domain-candidate · candidate





## 85. Original Rational-Expression Domain Guard

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. The original expression is E(x)=(8)/(x−(2)), excluding x=2. Ordered candidates=[2/1, 14/2, -9/1]. Give (three domain-eligibility values, three E values or undefined).

View answer and explanation([No, Yes, Yes], [undefined, 8/5, -8/11])

Check the original excluded point before evaluating; even a zero numerator cannot fill a denominator-zero hole or extend the original domain. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 8/5, -8/11]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0085/)

Variables and algebraic expressions · qf-density11-expression-domain-candidate · candidate





## 86. Original Rational-Expression Domain Guard

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. The original expression is E(x)=(9)/(x−(-2)), excluding x=-2. Ordered candidates=[-2/1, 7/2, -14/1]. Give (three domain-eligibility values, three E values or undefined).

View answer and explanation([No, Yes, Yes], [undefined, 18/11, -3/4])

Check the original excluded point before evaluating; even a zero numerator cannot fill a denominator-zero hole or extend the original domain. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 18/11, -3/4]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0086/)

Variables and algebraic expressions · qf-density11-expression-domain-candidate · candidate





## 87. Original Rational-Expression Domain Guard

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. The original expression is E(x)=(10)/(x−(-1)), excluding x=-1. Ordered candidates=[-1/1, 10/2, -14/1]. Give (three domain-eligibility values, three E values or undefined).

View answer and explanation([No, Yes, Yes], [undefined, 5/3, -10/13])

Check the original excluded point before evaluating; even a zero numerator cannot fill a denominator-zero hole or extend the original domain. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 5/3, -10/13]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0087/)

Variables and algebraic expressions · qf-density11-expression-domain-candidate · candidate





## 88. Original Rational-Expression Domain Guard

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. The original expression is E(x)=(11)/(x−(0)), excluding x=0. Ordered candidates=[0/1, 13/2, -14/1]. Give (three domain-eligibility values, three E values or undefined).

View answer and explanation([No, Yes, Yes], [undefined, 22/13, -11/14])

Check the original excluded point before evaluating; even a zero numerator cannot fill a denominator-zero hole or extend the original domain. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 22/13, -11/14]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0088/)

Variables and algebraic expressions · qf-density11-expression-domain-candidate · candidate





## 89. Original Rational-Expression Domain Guard

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. The original expression is E(x)=(12)/(x−(1)), excluding x=1. Ordered candidates=[1/1, 16/2, -14/1]. Give (three domain-eligibility values, three E values or undefined).

View answer and explanation([No, Yes, Yes], [undefined, 12/7, -4/5])

Check the original excluded point before evaluating; even a zero numerator cannot fill a denominator-zero hole or extend the original domain. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 12/7, -4/5]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0089/)

Variables and algebraic expressions · qf-density11-expression-domain-candidate · candidate





## 90. Original Rational-Expression Domain Guard

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. The original expression is E(x)=(13)/(x−(2)), excluding x=2. Ordered candidates=[2/1, 19/2, -14/1]. Give (three domain-eligibility values, three E values or undefined).

View answer and explanation([No, Yes, Yes], [undefined, 26/15, -13/16])

Check the original excluded point before evaluating; even a zero numerator cannot fill a denominator-zero hole or extend the original domain. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 26/15, -13/16]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0090/)

Variables and algebraic expressions · qf-density11-expression-domain-candidate · candidate





## 91. Original Rational-Expression Domain Guard

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. The original expression is E(x)=(14)/(x−(-2)), excluding x=-2. Ordered candidates=[-2/1, 12/2, -19/1]. Give (three domain-eligibility values, three E values or undefined).

View answer and explanation([No, Yes, Yes], [undefined, 7/4, -14/17])

Check the original excluded point before evaluating; even a zero numerator cannot fill a denominator-zero hole or extend the original domain. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 7/4, -14/17]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0091/)

Variables and algebraic expressions · qf-density11-expression-domain-candidate · candidate





## 92. Original Rational-Expression Domain Guard

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. The original expression is E(x)=(15)/(x−(-1)), excluding x=-1. Ordered candidates=[-1/1, 15/2, -19/1]. Give (three domain-eligibility values, three E values or undefined).

View answer and explanation([No, Yes, Yes], [undefined, 30/17, -5/6])

Check the original excluded point before evaluating; even a zero numerator cannot fill a denominator-zero hole or extend the original domain. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 30/17, -5/6]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0092/)

Variables and algebraic expressions · qf-density11-expression-domain-candidate · candidate





## 93. Original Rational-Expression Domain Guard

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. The original expression is E(x)=(16)/(x−(0)), excluding x=0. Ordered candidates=[0/1, 18/2, -19/1]. Give (three domain-eligibility values, three E values or undefined).

View answer and explanation([No, Yes, Yes], [undefined, 16/9, -16/19])

Check the original excluded point before evaluating; even a zero numerator cannot fill a denominator-zero hole or extend the original domain. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 16/9, -16/19]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0093/)

Variables and algebraic expressions · qf-density11-expression-domain-candidate · candidate





## 94. Original Rational-Expression Domain Guard

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. The original expression is E(x)=(17)/(x−(1)), excluding x=1. Ordered candidates=[1/1, 21/2, -19/1]. Give (three domain-eligibility values, three E values or undefined).

View answer and explanation([No, Yes, Yes], [undefined, 34/19, -17/20])

Check the original excluded point before evaluating; even a zero numerator cannot fill a denominator-zero hole or extend the original domain. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 34/19, -17/20]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0094/)

Variables and algebraic expressions · qf-density11-expression-domain-candidate · candidate





## 95. Original Rational-Expression Domain Guard

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. The original expression is E(x)=(18)/(x−(2)), excluding x=2. Ordered candidates=[2/1, 24/2, -19/1]. Give (three domain-eligibility values, three E values or undefined).

View answer and explanation([No, Yes, Yes], [undefined, 9/5, -6/7])

Check the original excluded point before evaluating; even a zero numerator cannot fill a denominator-zero hole or extend the original domain. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 9/5, -6/7]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0095/)

Variables and algebraic expressions · qf-density11-expression-domain-candidate · candidate





## 96. Original Rational-Expression Domain Guard

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. The original expression is E(x)=(19)/(x−(-2)), excluding x=-2. Ordered candidates=[-2/1, 17/2, -24/1]. Give (three domain-eligibility values, three E values or undefined).

View answer and explanation([No, Yes, Yes], [undefined, 38/21, -19/22])

Check the original excluded point before evaluating; even a zero numerator cannot fill a denominator-zero hole or extend the original domain. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 38/21, -19/22]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0096/)

Variables and algebraic expressions · qf-density11-expression-domain-candidate · candidate





## 97. Original Rational-Expression Domain Guard

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. The original expression is E(x)=(20)/(x−(-1)), excluding x=-1. Ordered candidates=[-1/1, 20/2, -24/1]. Give (three domain-eligibility values, three E values or undefined).

View answer and explanation([No, Yes, Yes], [undefined, 20/11, -20/23])

Check the original excluded point before evaluating; even a zero numerator cannot fill a denominator-zero hole or extend the original domain. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 20/11, -20/23]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0097/)

Variables and algebraic expressions · qf-density11-expression-domain-candidate · candidate





## 98. Original Rational-Expression Domain Guard

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. The original expression is E(x)=(21)/(x−(0)), excluding x=0. Ordered candidates=[0/1, 23/2, -24/1]. Give (three domain-eligibility values, three E values or undefined).

View answer and explanation([No, Yes, Yes], [undefined, 42/23, -7/8])

Check the original excluded point before evaluating; even a zero numerator cannot fill a denominator-zero hole or extend the original domain. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 42/23, -7/8]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0098/)

Variables and algebraic expressions · qf-density11-expression-domain-candidate · candidate





## 99. Original Rational-Expression Domain Guard

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. The original expression is E(x)=(22)/(x−(1)), excluding x=1. Ordered candidates=[1/1, 26/2, -24/1]. Give (three domain-eligibility values, three E values or undefined).

View answer and explanation([No, Yes, Yes], [undefined, 11/6, -22/25])

Check the original excluded point before evaluating; even a zero numerator cannot fill a denominator-zero hole or extend the original domain. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 11/6, -22/25]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0099/)

Variables and algebraic expressions · qf-density11-expression-domain-candidate · candidate





## 100. Original Rational-Expression Domain Guard

All operations and evaluations are exact in Q, not truncated natural-number arithmetic. The original expression is E(x)=(23)/(x−(2)), excluding x=2. Ordered candidates=[2/1, 29/2, -24/1]. Give (three domain-eligibility values, three E values or undefined).

View answer and explanation([No, Yes, Yes], [undefined, 46/25, -23/26])

Check the original excluded point before evaluating; even a zero numerator cannot fill a denominator-zero hole or extend the original domain. The exact result, retaining the requested order, is ([No, Yes, Yes], [undefined, 46/25, -23/26]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r03-variable-expression-0100/)

Variables and algebraic expressions · qf-density11-expression-domain-candidate · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261008-advance-20261011-r03-variable-expression-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261008-advance-20261011-r03-variable-expression-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261008-advance-20261011-r03-variable-expression-foundations/validation-report.json)