SMN01

Introduction to Algebra

1. Definitions of Key Terms

Here are essential terms in algebra:

• Algebra: A branch of mathematics using symbols to represent numbers and relationships.
• Variable: A symbol representing an unknown or changeable quantity, such as x or y.
• Coefficient: The number multiplying a variable, e.g., in 5x, 5 is the coefficient.
• Term: A single mathematical entity, like 3x or 7.
• Constant: A fixed number, like 3 in 2x + 3.
• Expression: A mathematical phrase without an equality sign, like 3x + 4.
• Equation: A statement of equality, like 2x + 5 = 11.
• Inequality: A comparison using symbols like < or >, e.g., x + 3 > 7.

2. Examples of Algebra in Daily Life

Finance: Calculating interest, discounts, or setting up budgets.

Cooking: Adjusting recipes by scaling ingredients (e.g., doubling the recipe).

Construction: Measuring and cutting materials to specific dimensions.

Travel: Calculating distances, time, and speed.

3. Collecting Terms: Like Terms

Like terms have the same variables raised to the same power and can be combined:

Example 1: 4x + 3x = 7x (Combine like terms with variable x).

Example 2: 2y - y = y.

Example 3: 5x² + 2x² = 7x² (Combine like terms with variable x²).

4. Collecting Terms: Unlike Terms

Unlike terms have different variables or powers and cannot be directly combined:

Example 1: In 3x + 4y, x and y are different variables.

Example 2: In 7x² + 3x, the powers of x differ (2 and 1), so they cannot be combined.

Example 3: In 2a + 5b - 3, there are different variables and a constant, which can’t be combined.

5. Numerical Values of an Expression

Substitute values to evaluate expressions:

Example 1: Evaluate 4x + 5 for x = 3. Solution: 4(3) + 5 = 12 + 5 = 17.

Example 2: Evaluate 2y - 7 for y = 4. Solution: 2(4) - 7 = 8 - 7 = 1.

Example 3: Evaluate 5x² for x = 2. Solution: 5(2)² = 5(4) = 20.

6. Simplification of Algebraic Expressions

Combine like terms to simplify expressions:

Example 1: Simplify 6a + 4a. Solution: 10a.

Example 2: Simplify 3x + 2x + x. Solution: 6x.

Example 3: Simplify 4y - 2y. Solution: 2y.

7. Simple Expansion

Distribute terms to expand an expression:

Example 1: Expand 2(x + 3). Solution: 2x + 6.

Example 2: Expand 3(y - 4). Solution: 3y - 12.

Example 3: Expand 5(x + 2y). Solution: 5x + 10y.

8. Expansion of Binomials

Expand two-term expressions:

Example 1: Expand (x + 2)(x + 4). Solution: x² + 6x + 8.

Example 2: Expand (x - 1)(x + 5). Solution: x² + 4x - 5.

Example 3: Expand (2x + 3)(x - 2). Solution: 2x² - x - 6.

9. Simple Factorization

Rewrite an expression as a product of simpler expressions:

Example 1: Factor 6x + 9. Solution: 3(2x + 3).

Example 2: Factor 10y - 5. Solution: 5(2y - 1).

Example 3: Factor 15a + 20b. Solution: 5(3a + 4b).

10. More on Factorization

Factor quadratics and differences of squares:

Example 1: Factor x² - 16. Solution: (x + 4)(x - 4).

Example 2: Factor 9a² - 4. Solution: (3a + 2)(3a - 2).

Example 3: Factor x² - 9x + 20. Solution: (x - 4)(x - 5).

11. Types of Algebraic Equations

Different types of equations include:

• Linear Equations: Forms like ax + b = c. Example: 2x + 3 = 7.
• Quadratic Equations: Equations with a power of 2, like x² + 5x + 6 = 0.
• Polynomial Equations: Higher powers, like x³ - 4x + 7 = 0.

12. Applications of Algebra

Algebra is widely used in various fields:

• Word Problems: Translating scenarios into equations.
• Predictive Modeling: Using equations to forecast outcomes.
• Physics and Engineering: Solving force, motion, and energy problems.