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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-20260924-equation-foundations · 2026-09-24 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
Over the integers, consider x+(-12)=-8 with candidate x=5. After substitution into the original equation, is the candidate a solution?
No
The left side is 5+(-12)=-7, while the right side is -8; the two sides are not equal.
Reviewed step-by-step derivation
Equation · qf-equation-solution-verification · candidate
Over the integers, consider x+(-11)=-7 with candidate x=4. After substitution into the original equation, is the candidate a solution?
Yes
The left side is 4+(-11)=-7, while the right side is -7; the two sides are equal.
Reviewed step-by-step derivation
Equation · qf-equation-solution-verification · candidate
Over the integers, consider x+(-10)=-6 with candidate x=5. After substitution into the original equation, is the candidate a solution?
No
The left side is 5+(-10)=-5, while the right side is -6; the two sides are not equal.
Reviewed step-by-step derivation
Equation · qf-equation-solution-verification · candidate
Over the integers, consider x+(-9)=-5 with candidate x=4. After substitution into the original equation, is the candidate a solution?
Yes
The left side is 4+(-9)=-5, while the right side is -5; the two sides are equal.
Reviewed step-by-step derivation
Equation · qf-equation-solution-verification · candidate
Over the integers, consider x+(-8)=-4 with candidate x=5. After substitution into the original equation, is the candidate a solution?
No
The left side is 5+(-8)=-3, while the right side is -4; the two sides are not equal.
Reviewed step-by-step derivation
Equation · qf-equation-solution-verification · candidate
Over the integers, consider x+(-7)=-3 with candidate x=4. After substitution into the original equation, is the candidate a solution?
Yes
The left side is 4+(-7)=-3, while the right side is -3; the two sides are equal.
Reviewed step-by-step derivation
Equation · qf-equation-solution-verification · candidate
Over the integers, consider x+(-6)=-2 with candidate x=5. After substitution into the original equation, is the candidate a solution?
No
The left side is 5+(-6)=-1, while the right side is -2; the two sides are not equal.
Reviewed step-by-step derivation
Equation · qf-equation-solution-verification · candidate
Over the integers, consider x+(-5)=-1 with candidate x=4. After substitution into the original equation, is the candidate a solution?
Yes
The left side is 4+(-5)=-1, while the right side is -1; the two sides are equal.
Reviewed step-by-step derivation
Equation · qf-equation-solution-verification · candidate
Over the integers, consider x+(-4)=0 with candidate x=5. After substitution into the original equation, is the candidate a solution?
No
The left side is 5+(-4)=1, while the right side is 0; the two sides are not equal.
Reviewed step-by-step derivation
Equation · qf-equation-solution-verification · candidate
Over the integers, consider x+(-3)=1 with candidate x=4. After substitution into the original equation, is the candidate a solution?
Yes
The left side is 4+(-3)=1, while the right side is 1; the two sides are equal.
Reviewed step-by-step derivation
Equation · qf-equation-solution-verification · candidate
Over the integers, consider x+(-2)=2 with candidate x=5. After substitution into the original equation, is the candidate a solution?
No
The left side is 5+(-2)=3, while the right side is 2; the two sides are not equal.
Reviewed step-by-step derivation
Equation · qf-equation-solution-verification · candidate
Over the integers, consider x+(-1)=3 with candidate x=4. After substitution into the original equation, is the candidate a solution?
Yes
The left side is 4+(-1)=3, while the right side is 3; the two sides are equal.
Reviewed step-by-step derivation
Equation · qf-equation-solution-verification · candidate
Over the integers, consider x+(0)=4 with candidate x=5. After substitution into the original equation, is the candidate a solution?
No
The left side is 5+(0)=5, while the right side is 4; the two sides are not equal.
Reviewed step-by-step derivation
Equation · qf-equation-solution-verification · candidate
Over the integers, consider x+(1)=5 with candidate x=4. After substitution into the original equation, is the candidate a solution?
Yes
The left side is 4+(1)=5, while the right side is 5; the two sides are equal.
Reviewed step-by-step derivation
Equation · qf-equation-solution-verification · candidate
Over the integers, consider x+(2)=6 with candidate x=5. After substitution into the original equation, is the candidate a solution?
No
The left side is 5+(2)=7, while the right side is 6; the two sides are not equal.
Reviewed step-by-step derivation
Equation · qf-equation-solution-verification · candidate
Over the integers, consider x+(3)=7 with candidate x=4. After substitution into the original equation, is the candidate a solution?
Yes
The left side is 4+(3)=7, while the right side is 7; the two sides are equal.
Reviewed step-by-step derivation
Equation · qf-equation-solution-verification · candidate
Over the integers, consider x+(4)=8 with candidate x=5. After substitution into the original equation, is the candidate a solution?
No
The left side is 5+(4)=9, while the right side is 8; the two sides are not equal.
Reviewed step-by-step derivation
Equation · qf-equation-solution-verification · candidate
Over the integers, consider x+(5)=-8 with candidate x=-13. After substitution into the original equation, is the candidate a solution?
Yes
The left side is -13+(5)=-8, while the right side is -8; the two sides are equal.
Reviewed step-by-step derivation
Equation · qf-equation-solution-verification · candidate
Over the integers, consider x+(6)=-7 with candidate x=-12. After substitution into the original equation, is the candidate a solution?
No
The left side is -12+(6)=-6, while the right side is -7; the two sides are not equal.
Reviewed step-by-step derivation
Equation · qf-equation-solution-verification · candidate
Over the integers, consider x+(7)=-6 with candidate x=-13. After substitution into the original equation, is the candidate a solution?
Yes
The left side is -13+(7)=-6, while the right side is -6; the two sides are equal.
Reviewed step-by-step derivation
Equation · qf-equation-solution-verification · candidate
Over the integers, consider x+(8)=-5 with candidate x=-12. After substitution into the original equation, is the candidate a solution?
No
The left side is -12+(8)=-4, while the right side is -5; the two sides are not equal.
Reviewed step-by-step derivation
Equation · qf-equation-solution-verification · candidate
Over the integers, consider x+(9)=-4 with candidate x=-13. After substitution into the original equation, is the candidate a solution?
Yes
The left side is -13+(9)=-4, while the right side is -4; the two sides are equal.
Reviewed step-by-step derivation
Equation · qf-equation-solution-verification · candidate
Over the integers, consider x+(10)=-3 with candidate x=-12. After substitution into the original equation, is the candidate a solution?
No
The left side is -12+(10)=-2, while the right side is -3; the two sides are not equal.
Reviewed step-by-step derivation
Equation · qf-equation-solution-verification · candidate
Over the integers, consider x+(11)=-2 with candidate x=-13. After substitution into the original equation, is the candidate a solution?
Yes
The left side is -13+(11)=-2, while the right side is -2; the two sides are equal.
Reviewed step-by-step derivation
Equation · qf-equation-solution-verification · candidate
Over the integers, consider x+(12)=-1 with candidate x=-12. After substitution into the original equation, is the candidate a solution?
No
The left side is -12+(12)=0, while the right side is -1; the two sides are not equal.
Reviewed step-by-step derivation
Equation · qf-equation-solution-verification · candidate
Solve x+(-15)=20 over the integers.
35
x=20−(-15)=35; substituting back satisfies the original equation.
Reviewed step-by-step derivation
Equation · qf-equation-additive-solve · candidate
Solve x+(-14)=19 over the integers.
33
x=19−(-14)=33; substituting back satisfies the original equation.
Reviewed step-by-step derivation
Equation · qf-equation-additive-solve · candidate
Solve x+(-13)=18 over the integers.
31
x=18−(-13)=31; substituting back satisfies the original equation.
Reviewed step-by-step derivation
Equation · qf-equation-additive-solve · candidate
Solve x+(-12)=17 over the integers.
29
x=17−(-12)=29; substituting back satisfies the original equation.
Reviewed step-by-step derivation
Equation · qf-equation-additive-solve · candidate
Solve x+(-11)=16 over the integers.
27
x=16−(-11)=27; substituting back satisfies the original equation.
Reviewed step-by-step derivation
Equation · qf-equation-additive-solve · candidate
Solve x+(-10)=15 over the integers.
25
x=15−(-10)=25; substituting back satisfies the original equation.
Reviewed step-by-step derivation
Equation · qf-equation-additive-solve · candidate
Solve x+(-9)=14 over the integers.
23
x=14−(-9)=23; substituting back satisfies the original equation.
Reviewed step-by-step derivation
Equation · qf-equation-additive-solve · candidate
Solve x+(-8)=13 over the integers.
21
x=13−(-8)=21; substituting back satisfies the original equation.
Reviewed step-by-step derivation
Equation · qf-equation-additive-solve · candidate
Solve x+(-7)=12 over the integers.
19
x=12−(-7)=19; substituting back satisfies the original equation.
Reviewed step-by-step derivation
Equation · qf-equation-additive-solve · candidate
Solve x+(-6)=11 over the integers.
17
x=11−(-6)=17; substituting back satisfies the original equation.
Reviewed step-by-step derivation
Equation · qf-equation-additive-solve · candidate
Solve x+(-5)=10 over the integers.
15
x=10−(-5)=15; substituting back satisfies the original equation.
Reviewed step-by-step derivation
Equation · qf-equation-additive-solve · candidate
Solve x+(-4)=9 over the integers.
13
x=9−(-4)=13; substituting back satisfies the original equation.
Reviewed step-by-step derivation
Equation · qf-equation-additive-solve · candidate
Solve x+(-3)=8 over the integers.
11
x=8−(-3)=11; substituting back satisfies the original equation.
Reviewed step-by-step derivation
Equation · qf-equation-additive-solve · candidate
Solve x+(-2)=7 over the integers.
9
x=7−(-2)=9; substituting back satisfies the original equation.
Reviewed step-by-step derivation
Equation · qf-equation-additive-solve · candidate
Solve x+(-1)=6 over the integers.
7
x=6−(-1)=7; substituting back satisfies the original equation.
Reviewed step-by-step derivation
Equation · qf-equation-additive-solve · candidate
Solve x+(0)=5 over the integers.
5
x=5−(0)=5; substituting back satisfies the original equation.
Reviewed step-by-step derivation
Equation · qf-equation-additive-solve · candidate
Solve x+(1)=4 over the integers.
3
x=4−(1)=3; substituting back satisfies the original equation.
Reviewed step-by-step derivation
Equation · qf-equation-additive-solve · candidate
Solve x+(2)=3 over the integers.
1
x=3−(2)=1; substituting back satisfies the original equation.
Reviewed step-by-step derivation
Equation · qf-equation-additive-solve · candidate
Solve x+(3)=2 over the integers.
-1
x=2−(3)=-1; substituting back satisfies the original equation.
Reviewed step-by-step derivation
Equation · qf-equation-additive-solve · candidate
Solve x+(4)=1 over the integers.
-3
x=1−(4)=-3; substituting back satisfies the original equation.
Reviewed step-by-step derivation
Equation · qf-equation-additive-solve · candidate
Solve x+(5)=0 over the integers.
-5
x=0−(5)=-5; substituting back satisfies the original equation.
Reviewed step-by-step derivation
Equation · qf-equation-additive-solve · candidate
Solve x+(6)=-1 over the integers.
-7
x=-1−(6)=-7; substituting back satisfies the original equation.
Reviewed step-by-step derivation
Equation · qf-equation-additive-solve · candidate
Solve x+(7)=-2 over the integers.
-9
x=-2−(7)=-9; substituting back satisfies the original equation.
Reviewed step-by-step derivation
Equation · qf-equation-additive-solve · candidate
Solve x+(8)=-3 over the integers.
-11
x=-3−(8)=-11; substituting back satisfies the original equation.
Reviewed step-by-step derivation
Equation · qf-equation-additive-solve · candidate
Solve x+(9)=-4 over the integers.
-13
x=-4−(9)=-13; substituting back satisfies the original equation.
Reviewed step-by-step derivation
Equation · qf-equation-additive-solve · candidate
Solve 1×x=-5 over the integers; the coefficient 1 is explicitly nonzero.
-5
1≠0, so x=-5/1=-5; substitution verifies the equality.
Reviewed step-by-step derivation
Equation · qf-equation-multiplicative-solve · candidate
Solve 2×x=-8 over the integers; the coefficient 2 is explicitly nonzero.
-4
2≠0, so x=-8/2=-4; substitution verifies the equality.
Reviewed step-by-step derivation
Equation · qf-equation-multiplicative-solve · candidate
Solve 3×x=-9 over the integers; the coefficient 3 is explicitly nonzero.
-3
3≠0, so x=-9/3=-3; substitution verifies the equality.
Reviewed step-by-step derivation
Equation · qf-equation-multiplicative-solve · candidate
Solve 4×x=-8 over the integers; the coefficient 4 is explicitly nonzero.
-2
4≠0, so x=-8/4=-2; substitution verifies the equality.
Reviewed step-by-step derivation
Equation · qf-equation-multiplicative-solve · candidate
Solve 5×x=-5 over the integers; the coefficient 5 is explicitly nonzero.
-1
5≠0, so x=-5/5=-1; substitution verifies the equality.
Reviewed step-by-step derivation
Equation · qf-equation-multiplicative-solve · candidate
Solve 6×x=0 over the integers; the coefficient 6 is explicitly nonzero.
0
6≠0, so x=0/6=0; substitution verifies the equality.
Reviewed step-by-step derivation
Equation · qf-equation-multiplicative-solve · candidate
Solve 7×x=7 over the integers; the coefficient 7 is explicitly nonzero.
1
7≠0, so x=7/7=1; substitution verifies the equality.
Reviewed step-by-step derivation
Equation · qf-equation-multiplicative-solve · candidate
Solve 8×x=16 over the integers; the coefficient 8 is explicitly nonzero.
2
8≠0, so x=16/8=2; substitution verifies the equality.
Reviewed step-by-step derivation
Equation · qf-equation-multiplicative-solve · candidate
Solve 9×x=27 over the integers; the coefficient 9 is explicitly nonzero.
3
9≠0, so x=27/9=3; substitution verifies the equality.
Reviewed step-by-step derivation
Equation · qf-equation-multiplicative-solve · candidate
Solve 1×x=4 over the integers; the coefficient 1 is explicitly nonzero.
4
1≠0, so x=4/1=4; substitution verifies the equality.
Reviewed step-by-step derivation
Equation · qf-equation-multiplicative-solve · candidate
Solve 2×x=10 over the integers; the coefficient 2 is explicitly nonzero.
5
2≠0, so x=10/2=5; substitution verifies the equality.
Reviewed step-by-step derivation
Equation · qf-equation-multiplicative-solve · candidate
Solve 3×x=-15 over the integers; the coefficient 3 is explicitly nonzero.
-5
3≠0, so x=-15/3=-5; substitution verifies the equality.
Reviewed step-by-step derivation
Equation · qf-equation-multiplicative-solve · candidate
Solve 4×x=-16 over the integers; the coefficient 4 is explicitly nonzero.
-4
4≠0, so x=-16/4=-4; substitution verifies the equality.
Reviewed step-by-step derivation
Equation · qf-equation-multiplicative-solve · candidate
Solve 5×x=-15 over the integers; the coefficient 5 is explicitly nonzero.
-3
5≠0, so x=-15/5=-3; substitution verifies the equality.
Reviewed step-by-step derivation
Equation · qf-equation-multiplicative-solve · candidate
Solve 6×x=-12 over the integers; the coefficient 6 is explicitly nonzero.
-2
6≠0, so x=-12/6=-2; substitution verifies the equality.
Reviewed step-by-step derivation
Equation · qf-equation-multiplicative-solve · candidate
Solve 7×x=-7 over the integers; the coefficient 7 is explicitly nonzero.
-1
7≠0, so x=-7/7=-1; substitution verifies the equality.
Reviewed step-by-step derivation
Equation · qf-equation-multiplicative-solve · candidate
Solve 8×x=0 over the integers; the coefficient 8 is explicitly nonzero.
0
8≠0, so x=0/8=0; substitution verifies the equality.
Reviewed step-by-step derivation
Equation · qf-equation-multiplicative-solve · candidate
Solve 9×x=9 over the integers; the coefficient 9 is explicitly nonzero.
1
9≠0, so x=9/9=1; substitution verifies the equality.
Reviewed step-by-step derivation
Equation · qf-equation-multiplicative-solve · candidate
Solve 1×x=2 over the integers; the coefficient 1 is explicitly nonzero.
2
1≠0, so x=2/1=2; substitution verifies the equality.
Reviewed step-by-step derivation
Equation · qf-equation-multiplicative-solve · candidate
Solve 2×x=6 over the integers; the coefficient 2 is explicitly nonzero.
3
2≠0, so x=6/2=3; substitution verifies the equality.
Reviewed step-by-step derivation
Equation · qf-equation-multiplicative-solve · candidate
Solve 3×x=12 over the integers; the coefficient 3 is explicitly nonzero.
4
3≠0, so x=12/3=4; substitution verifies the equality.
Reviewed step-by-step derivation
Equation · qf-equation-multiplicative-solve · candidate
Solve 4×x=20 over the integers; the coefficient 4 is explicitly nonzero.
5
4≠0, so x=20/4=5; substitution verifies the equality.
Reviewed step-by-step derivation
Equation · qf-equation-multiplicative-solve · candidate
Solve 5×x=-25 over the integers; the coefficient 5 is explicitly nonzero.
-5
5≠0, so x=-25/5=-5; substitution verifies the equality.
Reviewed step-by-step derivation
Equation · qf-equation-multiplicative-solve · candidate
Solve 6×x=-24 over the integers; the coefficient 6 is explicitly nonzero.
-4
6≠0, so x=-24/6=-4; substitution verifies the equality.
Reviewed step-by-step derivation
Equation · qf-equation-multiplicative-solve · candidate
Solve 7×x=-21 over the integers; the coefficient 7 is explicitly nonzero.
-3
7≠0, so x=-21/7=-3; substitution verifies the equality.
Reviewed step-by-step derivation
Equation · qf-equation-multiplicative-solve · candidate
Over the rational numbers, classify the solution set of 2x+(0)=5x+(-6).
unique solution x=2
(2−5)x=-6, so the equation has unique solution x=2.
Reviewed step-by-step derivation
Equation · qf-equation-linear-solution-class · candidate
Over the rational numbers, classify the solution set of 2x+(-3)=2x+(-3).
infinitely many solutions
Both sides reduce to the same expression, so every rational x is a solution.
Reviewed step-by-step derivation
Equation · qf-equation-linear-solution-class · candidate
Over the rational numbers, classify the solution set of 3x+(2)=3x+(3).
no solution
The variable terms cancel but the constants disagree, so there is no solution.
Reviewed step-by-step derivation
Equation · qf-equation-linear-solution-class · candidate
Over the rational numbers, classify the solution set of 5x+(3)=8x+(-3).
unique solution x=2
(5−8)x=-6, so the equation has unique solution x=2.
Reviewed step-by-step derivation
Equation · qf-equation-linear-solution-class · candidate
Over the rational numbers, classify the solution set of 5x+(0)=5x+(0).
infinitely many solutions
Both sides reduce to the same expression, so every rational x is a solution.
Reviewed step-by-step derivation
Equation · qf-equation-linear-solution-class · candidate
Over the rational numbers, classify the solution set of 6x+(5)=6x+(6).
no solution
The variable terms cancel but the constants disagree, so there is no solution.
Reviewed step-by-step derivation
Equation · qf-equation-linear-solution-class · candidate
Over the rational numbers, classify the solution set of 8x+(6)=11x+(0).
unique solution x=2
(8−11)x=-6, so the equation has unique solution x=2.
Reviewed step-by-step derivation
Equation · qf-equation-linear-solution-class · candidate
Over the rational numbers, classify the solution set of 8x+(3)=8x+(3).
infinitely many solutions
Both sides reduce to the same expression, so every rational x is a solution.
Reviewed step-by-step derivation
Equation · qf-equation-linear-solution-class · candidate
Over the rational numbers, classify the solution set of 9x+(8)=9x+(9).
no solution
The variable terms cancel but the constants disagree, so there is no solution.
Reviewed step-by-step derivation
Equation · qf-equation-linear-solution-class · candidate
Over the rational numbers, classify the solution set of 11x+(9)=14x+(3).
unique solution x=2
(11−14)x=-6, so the equation has unique solution x=2.
Reviewed step-by-step derivation
Equation · qf-equation-linear-solution-class · candidate
Over the rational numbers, classify the solution set of 11x+(6)=11x+(6).
infinitely many solutions
Both sides reduce to the same expression, so every rational x is a solution.
Reviewed step-by-step derivation
Equation · qf-equation-linear-solution-class · candidate
Over the rational numbers, classify the solution set of 12x+(11)=12x+(12).
no solution
The variable terms cancel but the constants disagree, so there is no solution.
Reviewed step-by-step derivation
Equation · qf-equation-linear-solution-class · candidate
Over the rational numbers, classify the solution set of 14x+(12)=17x+(6).
unique solution x=2
(14−17)x=-6, so the equation has unique solution x=2.
Reviewed step-by-step derivation
Equation · qf-equation-linear-solution-class · candidate
Over the rational numbers, classify the solution set of 14x+(9)=14x+(9).
infinitely many solutions
Both sides reduce to the same expression, so every rational x is a solution.
Reviewed step-by-step derivation
Equation · qf-equation-linear-solution-class · candidate
Over the rational numbers, classify the solution set of 15x+(14)=15x+(15).
no solution
The variable terms cancel but the constants disagree, so there is no solution.
Reviewed step-by-step derivation
Equation · qf-equation-linear-solution-class · candidate
Over the rational numbers, classify the solution set of 17x+(15)=20x+(9).
unique solution x=2
(17−20)x=-6, so the equation has unique solution x=2.
Reviewed step-by-step derivation
Equation · qf-equation-linear-solution-class · candidate
Over the rational numbers, classify the solution set of 17x+(12)=17x+(12).
infinitely many solutions
Both sides reduce to the same expression, so every rational x is a solution.
Reviewed step-by-step derivation
Equation · qf-equation-linear-solution-class · candidate
Over the rational numbers, classify the solution set of 18x+(17)=18x+(18).
no solution
The variable terms cancel but the constants disagree, so there is no solution.
Reviewed step-by-step derivation
Equation · qf-equation-linear-solution-class · candidate
Over the rational numbers, classify the solution set of 20x+(18)=23x+(12).
unique solution x=2
(20−23)x=-6, so the equation has unique solution x=2.
Reviewed step-by-step derivation
Equation · qf-equation-linear-solution-class · candidate
Over the rational numbers, classify the solution set of 20x+(15)=20x+(15).
infinitely many solutions
Both sides reduce to the same expression, so every rational x is a solution.
Reviewed step-by-step derivation
Equation · qf-equation-linear-solution-class · candidate
Over the rational numbers, classify the solution set of 21x+(20)=21x+(21).
no solution
The variable terms cancel but the constants disagree, so there is no solution.
Reviewed step-by-step derivation
Equation · qf-equation-linear-solution-class · candidate
Over the rational numbers, classify the solution set of 23x+(21)=26x+(15).
unique solution x=2
(23−26)x=-6, so the equation has unique solution x=2.
Reviewed step-by-step derivation
Equation · qf-equation-linear-solution-class · candidate
Over the rational numbers, classify the solution set of 23x+(18)=23x+(18).
infinitely many solutions
Both sides reduce to the same expression, so every rational x is a solution.
Reviewed step-by-step derivation
Equation · qf-equation-linear-solution-class · candidate
Over the rational numbers, classify the solution set of 24x+(23)=24x+(24).
no solution
The variable terms cancel but the constants disagree, so there is no solution.
Reviewed step-by-step derivation
Equation · qf-equation-linear-solution-class · candidate
Over the rational numbers, classify the solution set of 26x+(24)=29x+(18).
unique solution x=2
(26−29)x=-6, so the equation has unique solution x=2.
Reviewed step-by-step derivation
Equation · qf-equation-linear-solution-class · candidate
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