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Try an Example
Pick a scenario to see how the calculator works, then adjust the values
Simple 2x2 System
Solve 2x + 3y = 7 and x - y = 1 using Gaussian elimination
Key values: 2×2 system · Unique solution · x=2, y=1
3x3 Engineering System
Solve a 3-variable system commonly found in circuit analysis
Key values: 3×3 system · Cramer's Rule · 3 unknowns
Parallel Lines (No Solution)
Explore what happens when two equations have no intersection
Key values: 2×2 system · Inconsistent · No solution
This calculator is also known as Linear Equations Calculator.
Read the complete guideFrom Single Equations to Systems
A single linear equation (e.g., 3x + 2 = 8) has one unknown and one solution. A system of linear equations has multiple unknowns and requires as many independent equations as unknowns for a unique solution. The Rouche-Capelli theorem provides the classification framework.
Examples
Chemistry: Mixture Problem
Mix a 20% acid solution and a 50% acid solution to make 300 mL of 30% acid. How much of each?
x + y = 300 (total volume) and 0.2x + 0.5y = 90 (total acid). Solving: x = 200 mL of 20% solution, y = 100 mL of 50% solution.
Key takeaway: Mixture problems always produce a 2x2 system: one equation for quantity, one for concentration.
Understanding Linear Systems
Key concepts for linear equations:
- A system needs at least as many independent equations as unknowns for a unique solution
- The determinant tells you immediately whether a unique solution exists (det != 0)
- Rank analysis via Rouche-Capelli provides the full classification
- Parametric solutions express dependent variables in terms of free variables
Frequently Asked Questions about Linear Equations Calculator
What is the Rouche-Capelli theorem?
The Rouche-Capelli theorem states that a system Ax = b has a solution if and only if rank(A) = rank([A|b]). If rank(A) = rank([A|b]) = n (number of unknowns), the solution is unique. If rank(A) = rank([A|b]) < n, there are infinitely many solutions.
Why does the calculator show a condition number warning?
A high condition number (> 10^10) means the system is ill-conditioned: tiny changes in coefficients cause large changes in the solution. This is a numerical stability warning, not an error.
Can I solve systems with more than 3 equations?
This calculator supports 2x2 and 3x3 systems. For larger systems, consider a dedicated matrix calculator or a computer algebra system.
Specialized Calculators
Choose from 4 specialized versions of this calculator, each optimized for specific use cases and calculation methods.
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2 CalculatorsMethod Focus
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