What is a system of linear equations?
A system of linear equations is a set of two or more equations with the same variables. The goal is to find values that satisfy all equations simultaneously.
**2x2 system example:** \`\`\` 2x + y = 5 4x - y = 3 \`\`\` Solution: x = 1.333..., y = 2.333...
**3x3 system example:** \`\`\` 2x + y - z = 8 -3x - y + 2z = -11 -2x + y + 2z = -3 \`\`\` Solution: x = 2, y = 3, z = -1
How the solver works
This tool uses **Gaussian elimination with partial pivoting**, a numerically stable algorithm taught in linear algebra courses.
**Steps:** 1. Form the augmented matrix [A|b] from your coefficients 2. Apply forward elimination to create an upper triangular matrix 3. Use partial pivoting (swap rows) to avoid division by near-zero values 4. Back-substitute to find each variable's value
**Gaussian Elimination Formula:** \`\`\` R_i = R_i - (a_ij / a_jj) * R_j \`\`\` where R_i is the row being eliminated and R_j is the pivot row.
Types of solutions
A linear system can have exactly one of three outcomes:
- **Unique solution**: One specific set of values satisfies all equations. The matrix has full rank. - **No solution (inconsistent)**: The equations contradict each other. Example: x + y = 1 and x + y = 2. - **Infinite solutions (dependent)**: Equations are redundant - one is a multiple of another. The system is underdetermined.
Applications
Systems of linear equations appear throughout science, engineering, and economics:
- **Physics**: Solving circuit equations (Kirchhoff's laws) - **Economics**: Supply/demand equilibrium, input-output models - **Computer graphics**: Intersection of lines and planes - **Chemistry**: Balancing chemical equations - **Engineering**: Structural analysis, fluid flow networks - **Statistics**: Solving normal equations in least squares regression
How to use
1. Select 2x2 or 3x3 mode using the toggle 2. Enter all coefficients in the grid (including zeros) 3. The last column is the right-hand side constant 4. Click "Solve System" 5. View the solution values for x, y (and z) 6. Click "Show Steps" to see the full Gaussian elimination process
FAQs
Q: What does "no solution" mean? A: The equations are contradictory. For example, 2x + y = 5 and 2x + y = 7 cannot both be true simultaneously.
Q: What does "infinite solutions" mean? A: The equations are redundant - one is a linear combination of the others. There are infinitely many points that satisfy the system.
Q: Can I enter decimal or fractional coefficients? A: Yes. Enter any decimal number. The solver handles all real-number coefficients.
Q: What is partial pivoting? A: Before each elimination step, the algorithm selects the row with the largest absolute value in the pivot column and swaps it to the top. This prevents division by small numbers, which would amplify rounding errors.
Q: How does Gaussian elimination compare to substitution? A: Both methods give the same answer. Gaussian elimination is more systematic and easier to implement for larger systems. Substitution is more intuitive for small systems.
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