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Linear Equation Calculator

Solve linear equations in one variable, including identities and inconsistent cases. Compare coefficient decisions, share inputs, or export the full result.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all six published modes
Input interpretation
Enter values to calculate.
Result
—
Verified scope
all six published modes
Assurance
Core
Declared partition coverage
PASS · 7/7 declared partitions (invalid-domain, mode-solve, mode-inequality, mode-abs, mode-abs_inequality, mode-rational, mode-rational_inequality) · Matrix
Numerical scope
Listed independent full-pack vectors only; classification uses exact comparisons, no absolute epsilon.
Known limitations
  • Core CVP does not include live graph, viewport, or pointer interaction.
Model
x in ax + b = rhs.
Scope
Degree 1 only
Verification
Engine tested · Source checked · v1.5.1 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all six published modes· View Manifest · CVP overview · Specification
Versions
Calculation 1.5.1 · CVP protocol 1.0.0-proposed · Evidence 2026-10-28.meta-metrics-v2
Verification revision
2026-10-28.meta-metrics-v2 · 1/1 property · digest f9b7ec60bdd7
Legacy regression
18/18 tests · Production surface contract 6/6
Trust layers
Verification VERIFIED · Production CURRENT · Overall CVP state: VERIFIED under CVP 1.0.0-proposed
Reference
O1 model · O2 expected_values · O2 numerical_behavior
Interfaces
PASS · UI (SSR) / REST / MCP — ui-ssr is query-result HTML, not a live browser session.
Supplemental domain review
Not performed
Named expert review
Not performed
CVP suite
178/178 golden · 1/1 CVP boundary · 12/12 invalid · 1/1 property · 2/2 metamorphic · 178/178 cross-interface · 6/6 CVP contract · Manifest
Sources
Sources
Evidence
7 legacy golden · 3 legacy boundary · legacy regression suite · 178/178 oracle-backed golden · 12/12 invalid · Artifact integrity PASS
This calculator CURRENT · Public schema 1.5.1 matches · Semantic contract ✓ · Production attested · Public/cache ✓ · Origin ✓
Semantic contract
PASS
Full verification

Manifest identity, reference classes, interfaces, suite, and production records.

Formulas

Core equations used by this calculator.

Solutionx = (rhs − b) / a
Identitya = 0 and b = rhs → every x
Inconsistenta = 0 and b ≠ rhs → no solution
iThis page is not ax² + bx + c = 0. Use /calc/math/quadratic for quadratics. Linear inequalities are /linear-inequality. |ax+b|=rhs is /absolute-value.

How to use

1

Enter a, b, and rhs

b may be 0. rhs is the right-hand side (REST also accepts c).

2

Read x or the status

success, infinite, or inconsistent.

Example calculations

Common configurations with formula and result.

ϟ

One solution

2x + 3 = 11

x = (11 − 3) / 2
4
ϟ

Identity

0x + 5 = 5

every x
infinite

Linear Equation calculator specification

Version 1.5.1 · Engine tested

Calculation status

Review policy · Evidence

Definition
A linear equation in one unknown has the form ax + b = rhs. If a ≠ 0 there is exactly one solution x = (rhs − b)/a.
What it calculates
x in ax + b = rhs.
Inputs
  • a
  • b?
  • rhs
Outputs
  • x
  • status
Formula
x = (rhs − b) / a
Assumptions
  • Degree 1 only
  • Not quadratic
Units
  • dimensionless
Boundary conditions
  • missing a or rhs → MISSING_REQUIRED_INPUT
  • unknown mode → INVALID_MODE
Example
a=2 b=3 rhs=11 → x=4
Validation cases

2 published on this page · 18/18 tests · Production surface contract 6/6 · View evidence

  • a=2 b=3 rhs=11 → x=4
  • a=0 b=5 rhs=5 → status=infinite
Sources
  • ISO 80000-2:2019 Quantities and units — Mathematics
    Supports: Mathematical symbols and notation. This standard is not numerical verification or an endorsement of this implementation.
  • CRC Standard Mathematical Tables — linear equations, absolute value, and linear rational equations
    Supports: x = (rhs − b)/a; |ax+b| = rhs has two roots when rhs > 0; (ax+b)/(cx+d) = rhs clears to a linear equation excluding the pole
  • Margaret Stevens — The Maths Workbook (University of Oxford) — Chapter 1, §§1 and 3: Algebraic Expressions and Equations Chapter 1, §5: Inequalities and Absolute Value Chapter 1, §5: Inequalities and Absolute Value Chapter 1, §§1 and 3: Algebraic Expressions and Equations
    Supports: Algebraic fraction simplification and solving linear equations, including equations with parameters and rearranging formulas. A denominator must be nonzero; candidates introduced by clearing denominators must be checked in the original equation. Mathematical reference only; not implementation validation or an endorsement of CalculatorX. The piecewise definition of |x|, reversal of inequalities under a negative factor, and worked absolute-value equation and inequality solutions. The calculator applies these rules to its stated linear arguments and zero/negative right-hand-side cases. Mathematical reference only; not implementation validation or an endorsement of CalculatorX. The piecewise definition of |x|, reversal of inequalities under a negative factor, and worked absolute-value equation and inequality solutions. The calculator applies these rules to its stated linear arguments and zero/negative right-hand-side cases. Mathematical reference only; not implementation validation or an endorsement of CalculatorX. Algebraic fraction simplification and solving linear equations, including equations with parameters and rearranging formulas. A denominator must be nonzero; candidates introduced by clearing denominators must be checked in the original equation. Mathematical reference only; not implementation validation or an endorsement of CalculatorX.
  • University of Oxford Mathematical Institute — Algebra — Linear/quadratic inequalities and sign of factors
    Supports: Factoring polynomials and determining sign intervals, with explicit quadratic-inequality examples. For a rational inequality the calculator applies the same factor-sign rules to numerator and denominator separately and excludes denominator zeros; that rational extension is a derived application, not a claim that the page tests this solver. Mathematical reference only; not implementation validation or an endorsement of CalculatorX.
Calculation version
1.5.1

Background

Interpretation and common distinctions.

What does this linear equation calculator compute?

A linear equation in one unknown has the form ax + b = rhs. If a ≠ 0 there is exactly one solution x = (rhs − b)/a.

The published formula or method is x = (rhs − b) / a. The following references support the stated mathematics or method:

  • Margaret Stevens — The Maths Workbook (University of Oxford): Algebraic fraction simplification and solving linear equations, including equations with parameters and rearranging formulas. A denominator must be nonzero; candidates introduced by clearing denominators must be checked in the original equation. Mathematical reference only; not implementation validation or an endorsement of CalculatorX. The piecewise definition of |x|, reversal of inequalities under a negative factor, and worked absolute-value equation and inequality solutions. The calculator applies these rules to its stated linear arguments and zero/negative right-hand-side cases. Mathematical reference only; not implementation validation or an endorsement of CalculatorX. The piecewise definition of |x|, reversal of inequalities under a negative factor, and worked absolute-value equation and inequality solutions. The calculator applies these rules to its stated linear arguments and zero/negative right-hand-side cases. Mathematical reference only; not implementation validation or an endorsement of CalculatorX. Algebraic fraction simplification and solving linear equations, including equations with parameters and rearranging formulas. A denominator must be nonzero; candidates introduced by clearing denominators must be checked in the original equation. Mathematical reference only; not implementation validation or an endorsement of CalculatorX.
  • University of Oxford Mathematical Institute — Algebra: Factoring polynomials and determining sign intervals, with explicit quadratic-inequality examples. For a rational inequality the calculator applies the same factor-sign rules to numerator and denominator separately and excludes denominator zeros; that rational extension is a derived application, not a claim that the page tests this solver. Mathematical reference only; not implementation validation or an endorsement of CalculatorX.

Which inputs and limits should you check?

Follow the input domain, units, branch or solve direction, and rounding policy in the specification. Preserve exact outputs and any additional candidates or warnings.

The worked example uses 2x + 3 = 11 with x = (11 − 3) / 2. Compare it with the Example calculations and the complete result for the same inputs.

How can you verify and cite this result?

CalculatorX is responsible for this page. The mathematical references above explain the model; implementation checks and their scope are documented separately in the math.algebra.linear_equation v1.5.1 Evidence and capability contract. Read the displayed verification and production state together with the specification's numerical limits.

Use the clean calculator URL to cite the tool. A shared input link identifies a particular calculation. Content updates, numerical verification, and domain reviews have separate dates and scopes.

Numerical and input contract

Finite binary64 coefficients or strict decimal/scientific strings up to512 characters. Nonzero text rounding to zero rejects NUMERIC_UNDERFLOW. Exact rational decisions use the accepted binary64 coefficients, not arbitrary-precision raw decimals; no epsilon merges roots, zero coefficients, or poles. b defaults to0. In rational modes c and d form the denominator and cannot both be0. In other modes c/d are right-hand-side aliases and must agree with rhs.

The complete result includes solution ranges with open/closed endpoints. Endpoints round once to binary64. Overflow, nonzero underflow, and distinct exact boundaries collapsing to one binary64 value reject. Exact arithmetic is bounded to16384 numerator/denominator bits.178 independent Fraction predicate/open-cell full-pack references cover listed cases, with a1ULP fixed-vector ceiling and strict serialized topology; this is not a whole-domain/runtime proof. The figure may be omitted when boundaries cannot be separated faithfully; the numerical interval remains authoritative.

Typing calculates locally. Shared URLs and REST/MCP transmit inputs and may enter logs. CSV/JSON retain raw provided inputs and the complete unrounded result pack.

Other calculators in this family: Absolute Value Calculator, Absolute Value Inequality Calculator, Cubic Equation Calculator, Linear Inequality Calculator, Polynomial Evaluate Calculator, Polynomial Roots Calculator, Quadratic Formula Calculator, Quadratic Inequality Calculator . Explore all Algebra & Equations.

→ Absolute Value Calculator Solve absolute value equations in one variable. Find zero, one, two, or infinitely many solutions, preserve small roots, and share or export the full result. › → Absolute Value Inequality Calculator Solve absolute value inequalities with open or closed bounds. Inspect bounded intervals, unions, and degenerate cases, then share or export the solution set. › → Cubic Equation Calculator Solve real cubic equations with Cardano and trigonometric forms. Inspect all three roots, repeated roots, numerical limits, share inputs, and export results. › → Linear Inequality Calculator Solve linear inequalities with strict or inclusive bounds. Reverse the relation for negative coefficients, inspect the number line, and export the solution set. › → Polynomial Evaluate Calculator Evaluate real polynomials of degree 1–8 with exact rational Horner arithmetic and one binary64 rounding. Inspect numerical limits, share inputs, and export. › → Polynomial Roots Calculator Find all roots of real polynomials of degree 1–8, including complex and repeated roots. Check numerical limits, share inputs, and export complete results. › → Quadratic Formula Calculator Solve real quadratic equations with two, repeated, or complex roots. Inspect the discriminant, vertex, numerical limits, worked steps, and complete exports. › → Quadratic Inequality Calculator Find solution intervals for real quadratic inequalities. Choose strict or inclusive endpoints, inspect numerical limits, share raw inputs, and export results. ›

Frequently asked questions

Key distinctions behind the calculation.

Is this the quadratic formula?

No. Quadratics are /calc/math/quadratic. This page is degree 1.

Where is the inequality?

/calc/math/linear-inequality is mode=inequality on this same engine.

Where is absolute value?

/calc/math/absolute-value is mode=abs on this same engine: |ax + b| = rhs. The inequality is /absolute-value-inequality. (ax+b)/(cx+d)=rhs is /rational-equation. The inequality is /rational-inequality.

Where does this run?

Locally in the browser by default. REST and MCP call the same linear-equation engine.