Untangle the Algebra: Simplifying Expressions

Imagine tipping a laundry basket onto the floor: striped socks, plain socks, and odd mittens everywhere. You would never pair a striped sock with a mitten — they don't match. Algebra works exactly the same way. Before we simplify anything, we need to sort the pieces into matching groups. Here is your warm-up question:

An algebraic expression is a collection of terms joined by + or − signs. For example: 3x + 5y − x + 2y. A term is a single piece of an expression, such as 3x or 5y. The letter part of a term (x, y, x²) is called a variable — a symbol that stands for an unknown number. Like terms are terms that have exactly the same variable part. 3x and −x are like terms because both contain just x. 5y and 2y are like terms because both contain just y. But 3x and 5y are NOT like terms — x and y are different variables. Rule 1 — Collecting like terms: Add or subtract the coefficients (the numbers in front of the variables) of like terms, and keep the variable part unchanged. Think of it like the fruit example: you can add apples to apples. 3x + 5y − x + 2y = (3x − x) + (5y + 2y) ← group the like terms = 2x + 7y ← add the coefficients Rule 2 — Multiplying terms: When you multiply two terms together, multiply the coefficients and then multiply the variables using index rules. The index (or exponent) tells you how many times a variable is multiplied by itself: x² means x × x. Key index rule: x^m × x^n = x^(m+n) (add the powers when multiplying the same base) So: 3x² × 4x = (3 × 4) × (x² × x) = 12 × x^(2+1) = 12x³ Rule 3 — A common mistake to avoid: You can NEVER add unlike terms into one term. 2x + 7y cannot be written as 9xy — those are different quantities, just as apples and oranges stay separate.