Original question and givens
Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Ordered rows=[[2, -17], [-3, -16]] each contain [a,b] with a≠0. Give (root1, root2, does a common root exist?, common root or none). Use exact reduced fractions with positive denominators, retaining /1 for integers.
G001 — Over Q, ax+b=0 is linear only for a≠0; a=0 is a degenerate constant equation. Ordered rows=[[2, -17], [-3, -16]] each contain [a,b] with a≠0. Give (root1, root2, does a common root exist?, common root or none). Use exact reduced fractions with positive denominators, retaining /1 for integers.
Step-by-step derivation
S001 · implies
Before
source roles = [qf-density14-linear-two-row-coincidence, density14_linear_two_row_coincidence, finite_Q_affine_row_protocol]
After
derived = [[root1, root2, equal, common_root_or_none]]
Conditions: none
density14/15 use FULL-prefix underscores;density17 uses LOCAL hyphens.The coordinate and circle frames differ.Retain THIS family ordered answer roles/types and all original bilingual JSON/status.
Step identifiers and dependencies
Rule: R60_EXACT_perbank_operation_domain_FRAME_and_output_roles · Uses: G001
S002 · calculation
Before
Linear original row0 ORIGINAL negated constant := sub(0, -17)
After
Linear original row0 ORIGINAL negated constant = 17
Conditions: none
Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S001
S003 · calculation
Before
Linear original row0 ORIGINAL negative b over a := div(17, 2)
After
Linear original row0 ORIGINAL negative b over a = 17/2
Conditions: none
Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S002
S004 · calculation
Before
Linear original row0 reverse ORIGINAL a times root := mul(2, 17/2)
After
Linear original row0 reverse ORIGINAL a times root = 17
Conditions: none
Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S003
S005 · calculation
Before
Linear original row0 reverse ORIGINAL residual plus b := sum(17, -17)
After
Linear original row0 reverse ORIGINAL residual plus b = 0
Conditions: none
Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S004
S006 · implies
Before
source roles = [Linear original row0, 2, -17]
After
derived = [17/2, 0/1]
Conditions: none
ONLY nonzero a permits division:ax+b=0 IFF x=-b/a.Substitute into the ORIGINAL coefficients to verify zero residual.Normalize a negative a to a positive denominator;derived roots are not SAFE-capped or approximated.
Step identifiers and dependencies
Rule: Linear_NONZERO_a_exact_Q_root_AND_original_substitution · Uses: G001, S005
S007 · calculation
Before
Linear original row1 ORIGINAL negated constant := sub(0, -16)
After
Linear original row1 ORIGINAL negated constant = 16
Conditions: none
Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S006
S008 · calculation
Before
Linear original row1 ORIGINAL negative b over a := div(16, -3)
After
Linear original row1 ORIGINAL negative b over a = -16/3
Conditions: none
Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S007
S009 · calculation
Before
Linear original row1 reverse ORIGINAL a times root := mul(-3, -16/3)
After
Linear original row1 reverse ORIGINAL a times root = 16
Conditions: none
Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S008
S010 · calculation
Before
Linear original row1 reverse ORIGINAL residual plus b := sum(16, -16)
After
Linear original row1 reverse ORIGINAL residual plus b = 0
Conditions: none
Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S009
S011 · implies
Before
source roles = [Linear original row1, -3, -16]
After
derived = [-16/3, 0/1]
Conditions: none
ONLY nonzero a permits division:ax+b=0 IFF x=-b/a.Substitute into the ORIGINAL coefficients to verify zero residual.Normalize a negative a to a positive denominator;derived roots are not SAFE-capped or approximated.
Step identifiers and dependencies
Rule: Linear_NONZERO_a_exact_Q_root_AND_original_substitution · Uses: G001, S010
S012 · implies
Before
source roles = [[[2, -17], [-3, -16]]]
After
derived = [[17/2, -16/3], No, none]
Conditions: none
Require nonzero a in BOTH rows,solve each root independently,then compare exactly.Different coefficients can share a root.Without a common root return none,not numerical0 or approximate equality.
Step identifiers and dependencies
Rule: Linear_TWO_independent_nonzero_rows_root_coincidence_NOT_coefficients · Uses: G001, S011