Baseline-to-alternative trajectory through x-y decision spaceLive current inputs
Math & Statistics
Linear System Scenario Calculator
Compare how one fixed two-equation structure responds when the right-hand targets move from a baseline to an alternative scenario, with an adjustable blend and a visible solution trajectory.
Baseline solution-
Alternative solution-
Current blended solution-
Full-scenario change in x-
Full-scenario change in y-
Full solution displacement-
Displacement per 10% blend-
Shared determinant-
SOLUTION PATH
Baseline-to-alternative trajectory through x-y decision space
Because only the right-hand constants change, the solution follows a straight path. Milestones show 0%, 25%, 50%, 75%, and 100%; the live marker tracks the chosen blend.
SCENARIO MILESTONES
Exactly how the constants and solution move together
Each row solves the blended right-hand vector at a defined scenario weight.
| Scenario weight | Blended c | Blended f | Solved x | Solved y | Distance from baseline |
|---|
SCENARIO DESIGN
Change the decision drivers without silently changing the model
- Keep one coefficient matrix for both scenarios.
- Define baseline constants from a documented current case.
- Define alternative constants from one coherent scenario.
- Use the blend only for interpolation, not probability.
- Compare full displacement before focusing on the selected marker.
WHY THE PATH IS STRAIGHT
A fixed linear inverse maps a line segment to another line segment
Blending the constants does not blend two unrelated answers after the fact. It solves every intermediate right-hand vector with the same equations.
If coefficients also change, the path can curve and this fixed-matrix page is no longer the correct model.
PARAMETRIC SCENARIO MODEL
Interpolate the drivers first, then solve the shared system
The coefficient matrix stays fixed while the right-hand vector moves linearly. Matrix linearity makes the solution path linear as long as the determinant remains nonzero.
Detailed calculation process and general formulas
b(t) = (1 - t)b₀ + tb₁z(t) = A⁻¹b(t)Δz = A⁻¹(b₁ - b₀)z(t) = z₀ + tΔzdistance = √(Δx² + Δy²)Symbols, meanings, and units
- t
- alternative scenario blenddecimal from 0 to 1
- b₀,b₁
- baseline and alternative constant vectorsequation units
- z₀,z₁
- baseline and alternative [x,y] solutionssolution units
- Δz
- full scenario solution changesolution units
- A
- shared coefficient matrixcoefficient units
DECISION USE
Separate scenario magnitude from the selected commitment
The page makes both visible.
Endpoint change
-The alternative point shows the full consequence if the scenario is adopted.
Current blend
-The live marker shows the selected intermediate case.
Sensitivity
-Distance per ten percentage points makes scale comparable.
Decision takeaway: A blend is a planning interpolation, not evidence that the alternative is 60% likely.
SCENARIO RECORD
Keep these beside the comparison
- Source date for baseline constants
- Reason for alternative values
- Variables intentionally held fixed
- Feasible range of x and y
- Trigger for revisiting coefficients
- Owner of the scenario decision
Applied decisions
Two legitimate uses of one shared system
Capacity reallocation
Targets change while resource conversion coefficients stay fixed.
What the result clarifies: The trajectory reveals how both allocations move together.
Demand stress case
A higher first target and lower second target define one coherent alternative.
What the result clarifies: Endpoint displacement shows the consequence before a partial blend is chosen.
Worked default scenario
Current-input substitution and reconciliation
Method references
References for this calculator's specific method
Scope and limitations
This is a deterministic interpolation between two entered right-hand scenarios. It is not a probability forecast, optimization model, or justification for holding coefficients fixed when the real process changes.
Linear System Scenario Calculator | Baseline, Alternative, and Blended Solution FAQ
Is 60% blend a 60% probability?
No. It is simply 60% of the numeric movement from baseline constants to alternative constants.
Why is the trajectory straight?
The coefficient matrix is fixed, so its inverse is a linear transformation.
Can the coefficient matrix change too?
Not on this page; that requires solving separate systems and a different scenario model.
What if the determinant is zero?
The shared system has no unique solution, so the trajectory cannot be defined.