Introduce the basic concepts of algebra, including variables, constants, coefficients, and terms. Explore the use of algebraic expressions to represent real-world situations and mathematical relationships.
Introduction to Algebra – Study Notes
1. What Is Algebra?
- Definition: A branch of mathematics that uses symbols (letters, numbers, and operators) to represent quantities and relationships.
- Purpose: Allows us to describe patterns, solve unknowns, and model real‑world situations.
- Key Idea: Replace specific numbers with variables to create general formulas.
2. Basic Vocabulary
| Term |
Meaning |
| Variable |
Symbol (usually a letter) that stands for an unknown or changeable quantity (e.g., x, y). |
| Constant |
A fixed number that does not change (e.g., 3, –5, ½). |
| Coefficient |
The number multiplying a variable (e.g., in 3x, 3 is the coefficient). |
| Expression |
A combination of numbers, variables, and operations (e.g., 2x + 5). |
| Equation |
A statement that two expressions are equal (e.g., 2x + 5 = 13). |
| Term |
A single part of an expression separated by + or – (e.g., in 4x² – 7, the terms are 4x² and –7). |
| Exponent |
Indicates repeated multiplication of a base (e.g., x² = x·x). |
| Factor |
A number or expression that divides another exactly (e.g., 3 is a factor of 12). |
| Simplify |
Reduce an expression to its simplest form. |
3. Fundamental Operations
- Addition & Subtraction – Combine like terms (same variable and exponent).
Example: 3x + 5x = 8x
- Multiplication – Use the distributive property: a(b + c) = ab + ac.
Example: 2(x + 4) = 2x + 8
- Division – Divide each term by the divisor, often to isolate a variable.
Example: (6x)/3 = 2x
- Exponent Rules (for integer exponents)
- Product: aⁿ·aᵐ = aⁿ⁺ᵐ
- Quotient: aⁿ / aᵐ = aⁿ⁻ᵐ
- Power of a Power: (aⁿ)ᵐ = aⁿᵐ
4. Solving Linear Equations (One Variable)
General Steps
- Simplify each side – combine like terms, remove parentheses.
- Move variables to one side – add/subtract to isolate the variable term.
- Isolate the variable – divide or multiply to solve for the variable.
Example
Solve 2x + 7 = 19
| Step |
Operation |
Result |
| 1 |
Subtract 7 from both sides |
2x = 12 |
| 2 |
Divide both sides by 2 |
x = 6 |
5. Systems of Linear Equations (Two Variables)
- Goal: Find the ordered pair (x, y) that satisfies both equations.
- Common Methods
- Substitution – solve one equation for a variable, substitute into the other