Patterns and Algebra - Simplifying Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like $x$ or $y$), and operators (like $+$, $-$, $\times$, and $\div$). Visually, you can imagine variables as empty boxes or containers where different numbers can be placed.

A term is a single part of an expression separated by plus or minus signs. For example, in the expression $5x + 3y - 7$, there are three terms: $5x$, $3y$, and $-7$. A constant is a term that is just a number without a variable.

The coefficient is the numerical factor of a term containing a variable. In the term $4x$, the number $4$ is the coefficient. If a variable appears alone, like $z$, its coefficient is understood to be $1$.

Like terms are terms that have the exact same variable parts raised to the same power. Visually, you can think of like terms as items of the same category, such as apples and oranges; you can add $3$ apples and $2$ apples to get $5$ apples, but you cannot combine $3$ apples and $2$ oranges.

Simplifying an expression involves combining like terms by adding or subtracting their coefficients. For example, $5a + 2a$ simplifies to $7a$. Visualizing this on a number line can help when dealing with negative coefficients, such as $4x - 6x = -2x$.

The Distributive Property allows you to remove parentheses by multiplying the term outside the bracket by every term inside. It is written as $a(b + c) = ab + ac$. Visually, imagine drawing arrows from the outside term to each term inside the parentheses to ensure nothing is missed.

When simplifying complex expressions, follow the order of operations (BODMAS/PEMDAS). This means dealing with Brackets first, then multiplication/division from left to right, and finally addition/subtraction from left to right.