About Lesson
| Method | Typical Equation | Steps to Resolve | Limitations | Benefits |
|---|---|---|---|---|
| Graphical Method | y = mx + c | Plot each equation and find intersection points. | Impractical for more than 2 variables; accuracy depends on scale. | Intuitive and visual; good for understanding the nature of solutions. |
| Substitution Method | x + y = b | Solve one equation for a variable, substitute it into others, and solve. | Can be cumbersome for complex systems. | Simple and straightforward for small systems. |
| Elimination Method | ax + by = c | Add or subtract equations to eliminate a variable, then solve for others. | Can get complex with many variables. | Effective for linear equations; straightforward for small systems. |
| Matrix Method (Inversion) | Ax = B | Formulate matrix equation, calculate inverse of A, compute A-1B. | Infeasible for non-square or singular matrices. | Systematic and precise; good for complex systems. |
| Gaussian Elimination | Ax = B | Convert to upper triangular form using row operations, then back substitute. | Can be computationally intensive for large matrices. | General method, applicable to most systems. |
| Gauss-Jordan Elimination | Ax = B | Reduce matrix to row echelon form, directly read off solutions. | Similar to Gaussian; can be computationally intensive. | Simplifies to a direct solution without back substitution. |
| LU Decomposition | Ax = B | Decompose A into LU, solve Ly = B and then Ux = y. | Requires additional steps to perform decomposition. | Efficient for multiple systems with the same A. |
| Singular Value Decomposition | Ax = B | Decompose A into U, Σ, V, use these to solve the system. | Complex and requires understanding of advanced linear algebra. | Powerful in data science and for ill-conditioned systems. |
| Iterative Methods | Ax = B | Start with a guess, iteratively refine the solution. | Convergence can be slow; not always guaranteed. | Useful for very large systems where direct methods fail. |
| Cramer’s Rule | ax + by = c | Use determinants to solve, each variable calculated separately. | Only for square matrices with non-zero determinants. | Straightforward for small systems; provides direct solution. |