1. What is an Equation?
- Definition: A mathematical statement showing a relationship between two expressions using symbols like =, <, >,, or.
- Types:
- Simple equations: Involve numbers only (e.g., (2 + 2 = 4)).
- Algebraic equations: Include letters (variables) to represent unknown numbers (e.g., (2 + x = 4)).
- Purpose: Solve equations to find the value of the variable (e.g., (x = 4)).
2. Variables and Constants
- Variable: A letter (e.g., (x)) representing different values in equations.
- Constant: A value that doesn’t change (e.g., (\pi = 3.142)).
- Example:
- In (2x), (x) is the variable, and 2 is the coefficient (multiplier).
3. Terms in an Equation
- Definition: Individual parts of an equation separated by (+) or (-).
- Like Terms: Terms with the same variables (e.g., (2x + 3x = 5x)).
- Unlike Terms: Terms with different variables or combinations (e.g., (2x + 3y)).
4. Solving Equations
- Rearrange equations to isolate the variable (e.g., (x)).
- Steps:
- Follow the rules of balance: Whatever is done to one side must also be done to the other.
- Use BODMAS (Brackets, Orders, Division/Multiplication, Addition/Subtraction).
- Simplify by combining like terms and removing fractions.
Examples:
5. Equations and Graphs
- Graphing: Visualize equations by plotting variables (x) (horizontal axis) and (y) (vertical axis).
- Linear Equation: Straight-line graph (e.g., (y = 2x + 3)).
- For (x = 0), (y = 3). For (x = 1), (y = 5).
- Quadratic Equation: Parabolic graph (e.g., (y = x^2 + x + 4)).
Advantages:
- Find values: Determine (y) for given (x) (or vice versa).
- Extrapolation: Estimate beyond known values by extending the graph.
6. Key Takeaways
- Algebra simplifies and generalizes problem-solving.
- Practice isolating variables, solving equations, and plotting graphs.
- Next steps: Tackle advanced concepts like simultaneous and quadratic equations.
Tip: Break equations into manageable parts to reduce complexity.