Algebra - Addition and Subtraction of Algebraic Expressions

Algebraic Terms and Coefficients: An algebraic expression is made of terms. A term is a product of a number (the numerical coefficient) and variables (literal factors). For example, in the term $-5x^{2}y$, $-5$ is the coefficient and $x^{2}y$ is the literal part. Visually, you can imagine a term as a single 'package' where the number is the quantity and the variables are the type of item.

Like and Unlike Terms: Like terms are terms that have the exact same literal factors with the same exponents. For instance, $3ab$ and $-7ab$ are like terms, while $3a$ and $3b$ are unlike. Visually, think of like terms as identical fruits (like apples); you can only combine items if they are the same 'fruit' type.

Addition of Algebraic Expressions: To add expressions, we only combine the numerical coefficients of the like terms while keeping the literal part unchanged. For example, $2x + 3x = 5x$. In your mind, visualize grouping all identical variable 'shapes' together on a desk before counting the total number for each shape.

Subtraction and the Additive Inverse: Subtracting an algebraic expression is the same as adding its additive inverse. To find the additive inverse, change the sign of every term within the expression. Visually, imagine a 'sign-flipper' gate: as terms like $(2x - 3y)$ pass through a subtraction gate, they transform into $-2x + 3y$.

The Horizontal Method: This method involves writing all expressions in a single horizontal line. You use brackets to separate expressions and then rearrange the terms so that like terms are adjacent to each other. Visually, this looks like a long string of terms that gradually shrinks as you 'shrink' groups of like terms into single terms.

The Column Method: In this method, expressions are written in rows, one below the other. You must align the terms such that like terms fall in the same vertical column (e.g., all $x$ terms in one column, all $y$ terms in another). Visually, this looks like a standard multi-digit addition or subtraction problem, where each column represents a specific variable 'category'.

Rules for Brackets (BODMAS): When expressions involve brackets, always simplify the terms inside the innermost brackets first. If a bracket is preceded by a minus sign, every term inside the bracket must have its sign changed when the bracket is removed. For example, $5 - (2x + 3) = 5 - 2x - 3$.

Combining Multiple Terms: When an expression has many terms, the final result is reached when no more like terms remain. The result is often arranged in descending powers of the variable (e.g., $x^{2}$ then $x$ then constants) to keep it visually organized and easy to read.