Algebraic Expressions - Addition and Subtraction of Algebraic Expressions

Algebraic expressions are formed by combining variables and constants using arithmetic operations like addition and subtraction. Visually, think of an expression like $3x + 5$ as a combination of three unknown 'boxes' ($x$) and five fixed 'units' ($5$).

Terms are the individual parts of an expression separated by $+$ or $-$ signs. For example, in $5x^2 - 3xy$, the terms are $5x^2$ and $-3xy$. You can visualize this as a tree structure where the expression is the trunk and each term is a major branch.

A factor is a multiplier within a term. In the term $4xy$, the factors are $4, x, \text{ and } y$. The numerical factor is called the coefficient. For example, in $-7ab$, the coefficient is $-7$. Visualizing a term as a product box helps identify these components.

Like terms are terms that contain the same variables raised to the same powers, regardless of their numerical coefficients. For example, $2x^2y$ and $-5x^2y$ are like terms. Unlike terms have different variables or powers, such as $3x$ and $3x^2$. Addition and subtraction can only be performed on like terms.

To add algebraic expressions, we identify and group the like terms together, then add their numerical coefficients. Visually, this is like sorting different types of fruit; you can add 3 apples to 2 apples to get 5 apples, but you cannot combine apples and oranges into a single unit.

To subtract one expression from another, we add the additive inverse of the expression being subtracted. This means changing the sign of every term in the expression that follows the minus sign (from $+$ to $-$ and from $-$ to $+$).

There are two main layouts for performing these operations: the Horizontal Method, where terms are grouped side-by-side in a single line, and the Column Method, where expressions are written one below the other with like terms aligned in vertical columns for easy calculation.