# Problem-Solving Methods in University Mathematics
## MTH Student Academy

### 1. The Structured Approach
When facing a university-level mathematics problem, follow this four-step method:

1.  **Analyze**: Identify what is given and what is required. Draw a diagram if possible.
2.  **Translate**: Convert the word problem or scenario into mathematical symbols and equations.
3.  **Execute**: Apply the relevant theorems, rules, and algebraic steps to solve the equations.
4.  **Verify**: Check if the answer makes physical or logical sense. Plug the result back into the original equation.

### 2. Common Techniques
- **Substitution**: Simplify complex expressions by replacing terms with a single variable (e.g., u-substitution in calculus).
- **Symmetry**: Look for even/odd functions or geometric symmetry to reduce calculation time.
- **Limiting Cases**: Test your solution with extreme values (0, infinity) to see if it behaves as expected.
- **Dimensional Analysis**: Ensure that the units on both sides of your equation match.

### 3. Proof Strategies
For advanced courses like Discrete Mathematics or Linear Algebra:
- **Direct Proof**: Assume P is true and show that Q must follow.
- **Proof by Contradiction**: Assume the statement is false and show that this leads to a logical impossibility.
- **Mathematical Induction**: Prove for a base case (n=1) and show that if it holds for 'k', it must hold for 'k+1'.