Augmented matrix descending to the solved pairLive current inputs
Math & Statistics
Linear System Step-by-Step Calculator
Solve a two-equation linear system while exposing the determinant classification, the exact elimination operation, both unknowns, and two substitution residuals instead of returning an unexplained ordered pair.
System classification-
Coefficient determinant-
Solved x-
Solved y-
Elimination multiplier-
Equation 1 residual-
Equation 2 residual-
Independent substitution check-
ROW-OPERATION STORYBOARD
Augmented matrix descending to the solved pair
The first matrix preserves the entered equations, the middle panel applies one visible row operation, and the final panel isolates the second variable before back-substitution.
ARITHMETIC AUDIT
Every determinant, numerator, and residual in one ledger
The register makes the two solution paths independently inspectable.
| Quantity | Symbolic form | Substitution | Value | Interpretation |
|---|
EQUATION ENTRY
Preserve signs and the same variable order
- Write both equations as ax + by = c.
- Enter an explicit zero for a missing variable.
- Keep x first and y second in both rows.
- Retain negative signs on coefficients and constants.
- Use the residual rows to catch transcription or rounding errors.
WHY THE DETERMINANT COMES FIRST
Parallel and coincident lines cannot be solved by ordinary division
A nonzero determinant means the coefficient rows point in different directions, so the two lines meet once. A zero determinant requires a consistency test on the augmented constants.
The elimination multiplier is not a new assumption. It is the exact factor that cancels x from the second row, leaving one equation in y.
EXACT TWO-BY-TWO SOLUTION
Classify before dividing by the determinant
The coefficient determinant decides whether a unique ordered pair exists. When it is nonzero, Cramer's Rule and elimination must agree; the final residuals test the entered equations directly.
Detailed calculation process and general formulas
Δ = ae - bdΔx = ce - bfΔy = af - cdx = Δx / Δy = Δy / ΔSymbols, meanings, and units
- a,b,d,e
- coefficients multiplying x and ycoefficient units
- c,f
- right-hand constantsequation units
- Δ
- coefficient determinantcoefficient²
- Δx, Δy
- replacement determinantsmixed equation units
- r₁,r₂
- left side minus right side after substitutionequation units
THREE-WAY RECONCILIATION
A trustworthy answer survives three independent views
The page links geometry, determinants, and substitution.
Geometry
-The classification states whether the entered lines intersect, coincide, or remain parallel.
Determinants
-Replacement determinants produce the ordered pair without hiding the division.
Residuals
-Both entered equations are recomputed from the displayed solution.
Decision takeaway: Do not report x and y when the determinant is zero; report the system classification instead.
COMMON FAILURE POINTS
Check these before blaming the algebra
- Swapped coefficient order
- Lost negative sign
- Rounded intermediate determinant
- Constant placed on the wrong side
- Division attempted when Δ = 0
Applied decisions
Two systems that look similar but behave differently
Unique intersection
The rows have different coefficient ratios, so elimination leaves a nonzero pivot.
What the result clarifies: Both equations identify one ordered pair and both residuals close to zero.
Dependent or inconsistent
The coefficient rows are proportional.
What the result clarifies: The constants decide whether the lines coincide or remain parallel.
Worked default scenario
Current-input substitution and reconciliation
Method references
References for this calculator's specific method
Scope and limitations
This calculator solves a two-variable linear algebra model exactly to floating-point precision. It does not determine whether the equations are appropriate for a real process, and very small determinants can magnify measurement and rounding error.
Linear System Step-by-Step Calculator | Elimination, Determinants, and Check FAQ
Why can the determinant be zero?
The coefficient rows are linearly dependent, so the lines have the same direction and do not define one unique intersection.
What residual is acceptable?
For exact entered numbers it should be near floating-point zero; measurement-based models may need a domain-specific tolerance.
Can I enter decimals?
Yes. Keep enough precision and inspect the determinant before rounding the answer.
Does this solve more than two equations?
No. This page is intentionally limited to a two-by-two system.