Rational Equation Calculator
Solve (ax + b)/(cx + d) = rhs. Same engine as Linear Equation. Clears to linear; the pole x = −d/c is excluded. Not quadratic. Not a second engine. Not CAS. Runs locally.
Trust summary Engine tested · Specification checked · 18/18 tests · Production surface contract 4/4 · v1.5.0
- Input interpretation
- Enter values to calculate.
- Result
- —
- Model
- The real solutions of (ax + b)/(cx + d) = rhs, excluding the pole.
- Scope
- Degree 1 numerator and denominator
- Verification
- Engine tested · 18/18 tests · Production surface contract 4/4 · Specification checked · v1.5.0
- Named expert review
- Optional · Not performed
- Specification basis
- ISO 80000-2:2019 Quantities and units — Mathematics
- Evidence
- 7 golden · 3 boundary · 8 property · Production surface contract 4/4 · Artifact integrity PASS
- Production
- Embedded snapshot: STALE · Last attested schema matched 1.5.0 snapshot / local build · Semantic contract ✓ · Last attestation PASS · current evidence changed · re-attestation required · Public/cache ✓ · Origin ✓ · Live production status STALE (2 capabilities; 162 remain CURRENT) @ 2026-09-20T09:01:12.147Z
- Semantic contract
- PASS
Formulas
Core equations used by this calculator.
How to use
Enter a, b, c, d, and rhs
(2x + 3)/(x + 1) = 1 is a=2, b=3, c=1, d=1, rhs=1. The linear page stays at /linear-equation.
Read x or the excluded pole
x = −2. The pole x = −1 is never a solution. Identity except the pole is ℝ \ {pole}.
Example calculations
Common configurations with formula and result.
One root
(2x + 3)/(x + 1) = 1
Identity except pole
(2x + 2)/(x + 1) = 2
Empty
(2x + 3)/(x + 1) = 2
Rational Equation calculator specification
Version 1.5.0 · Engine tested
- Engine tested 18/18 tests · Production surface contract 4/4
- Named expert review Not performed
- Calculation version 1.5.0
- Definition
- (ax + b)/(cx + d) = rhs clears to (a − rhs·c)x = rhs·d − b. This page is mode=rational on math.algebra.linear_equation. The pole x = −d/c is never a solution. Identically 0 denominator is invalid. Not a second engine. Not quadratic. Not CAS.
- What it calculates
- The real solutions of (ax + b)/(cx + d) = rhs, excluding the pole.
- Inputs
- a
- b?
- c?
- d?
- rhs
- Outputs
- x
- pole
- status
- n
- Formula
x = (rhs·d − b) / (a − rhs·c), x ≠ −d/c- Assumptions
- Degree 1 numerator and denominator
- Real coefficients
- Not quadratic
- Not an inequality — use /rational-inequality
- Not CAS
- Units
- dimensionless
- Boundary conditions
- missing a or rhs → MISSING_REQUIRED_INPUT
- c = d = 0 → INVALID_INPUT
- unknown mode (including cas, quadratic) → INVALID_MODE
- root equals pole → empty
- Example
- a=2 b=3 c=1 d=1 rhs=1 → x=-2
- Validation cases
2 published on this page · 18/18 tests · Production surface contract 4/4 · View evidence
- a=2 b=3 c=1 d=1 rhs=1 → x=-2
- mode=cas a=2 b=3 c=1 d=1 rhs=1 → INVALID_MODE
- Specification basis
- ISO 80000-2:2019 Quantities and units — Mathematics
- Calculation version
- 1.5.0
Related tools
Other calculators in this family: Absolute Value Calculator, Absolute Value Inequality Calculator, Column Space Calculator, Condition Number Calculator, Cubic Equation Calculator, Least Squares Calculator, Left Nullspace Calculator, Linear Equation Calculator . Explore all Algebra & Equations.
Frequently asked questions
Key distinctions behind the calculation.
Is this a second linear engine?
No. It is mode=rational on math.algebra.linear_equation, the same engine as /calc/math/linear-equation.
Why is the pole excluded?
cx + d = 0 makes the original fraction undefined, even if clearing the denominator produces that x.
Do you solve rational inequalities?
/calc/math/rational-inequality is mode=rational_inequality on this same engine.