Introduction to Algebra

What is Algebra?

Algebra is similar to arithmetic but introduces the element of the unknown.
Follows the same four operations of arithmetic: addition, subtraction, multiplication, and division.

In arithmetic, the unknown is typically the answer to a problem (e.g., 1 + 2 = ?).
In algebra, an unknown is represented by a symbol, usually a letter (e.g., 1 + 2 = x).
This symbol acts as a placeholder for the unknown value, which we aim to solve.

An equation is a mathematical statement that asserts the equality of two expressions.
Example: 1 + 2 = x. Here, 1 + 2 on one side equals x on the other side.
Solving the equation involves finding the value of the unknown that makes the equation true.
Equations can be simple or complex (e.g., x - 2 = 1).

Same symbol/letter can stand for different unknown values in different problems.
Example: x in 1 + 2 = x, and x in 5 + x = 10.
Within the same problem, the same symbol must always represent the same value.
Different symbols can represent the same number (e.g., a + b = 2).
Symbols whose values can change are called variables.
Example: If a is 0, b is 2; if a is 1, b is 1; if a is 2, b is 0.

Multiplication is the default operation in Algebra.
If no operation is shown between two symbols, assume they are being multiplied (e.g., ab means a * b).
Simplifies notation (e.g., 2x means 2 * x).
Known numbers still require explicit multiplication symbols (e.g., 2 * 5 still needs a symbol or use parentheses).

Used to model and describe real-world phenomena.
Graphing solutions of equations can help visualize and predict real-life scenarios.
Linear equations form straight lines and can describe slopes and travel time.
Quadratic equations can design lenses, describe projectile motion, and predict population growth.
Crucial in fields like science, engineering, economics, and computer programming.
Even though not always needed daily, algebra is a useful part of math.