Introduction to Algebra

What is Algebra?

• Algebra is a branch of math similar to arithmetic.
• Uses the four main arithmetic operations: addition, subtraction, multiplication, and division.
• Introduces the concept of the unknown (usually represented by a letter like 'x').

Key Concepts

Equations

• An equation is a mathematical statement that asserts the equality of two expressions.
• Example: 1 + 2 = x (In this case, x = 3)
• Aim in algebra: Solve for the unknown values in equations.

Using Symbols

• Symbols (letters) stand in for unknown numbers.
• Example: In 5 + x = 10, x stands for 5.
• A symbol can’t stand for different values within the same problem.
• Example: x + x = 10 could not mean 6 + 4.
• Different symbols can represent the same number.
• Example: a + b = 2 (a = 1, b = 1).

Variables

• A variable's value can change and is thus called a "variable" (e.g., a and b in the example above).
• We use the term "variable" for any letter in an equation representing an unknown, especially when its value can change.

Multiplication in Algebra

• Multiplication is the default operation in algebra. If no operation is shown, multiplication is assumed.
• Example: Instead of "a * b", you can write "ab".
• The multiplication symbol is still needed between two known numbers.
• Parentheses can also imply multiplication. For example, (a + b)(x + y).
• Parentheses can help avoid confusion in specific contexts. E.g., 2 x 5 can be written as 2(5) to avoid looking like 25.

Real-World Application of Algebra

• Algebra is useful for modeling real-world situations.
• Graphing solutions from algebraic equations can help describe and predict real-life scenarios.
• Linear Equations: Form straight lines; used for describing slopes or predicting travel times.
• Quadratic Equations: Form curves; used in designing lenses, predicting projectile paths, or population growth.
• Algebra is widely used in science, engineering, economics, and computer programming.