Introduction to Algebra
Algebra is the branch of mathematics that deals with symbols and the rules for manipulating these symbols. It is a unifying thread of almost all of mathematics and has applications in virtually every field.
Variables and Constants
In algebra, we use letters (like $x$, $y$, and $z$) to represent variables — quantities that can change or take different values. Constants, on the other hand, are fixed values (like $2$, $\pi$, or $e$).
An algebraic expression combines variables and constants with operations like addition, subtraction, multiplication, and division.
Basic Operations
Let’s review the basic algebraic operations:
Addition and Subtraction: $x + y$ represents the sum of $x$ and $y$ $x - y$ represents the difference of $x$ and $y$
Multiplication: $x \times y$ or simply $xy$ represents the product of $x$ and $y$
Division: $\frac{x}{y}$ represents $x$ divided by $y$ (where $y \neq 0$)
Example: Simplifying Expressions
Let’s simplify the expression: $3x + 2y - 5x + 7y$
- Group like terms: $(3x - 5x) + (2y + 7y)$
- Combine like terms: $-2x + 9y$
Algebraic Properties
Some important properties to remember:
Commutative Property: $a + b = b + a$ $a \times b = b \times a$
Associative Property: $(a + b) + c = a + (b + c)$ $(a \times b) \times c = a \times (b \times c)$
Distributive Property: $a(b + c) = ab + ac$
Practice Problems
Try simplifying these expressions:
- $4x + 3y - 2x + y$
- $2(3x - 4) + 5x$
- $\frac{x^2 - 4}{x - 2}$ (where $x \neq 2$)
Solutions will be discussed in the next chapter.