Foundations — Addition and subtraction of algebraic expressions
Before you can add or subtract algebraic expressions, you need to read them. This page takes every symbol the parent note used — , , , , the , the , the parentheses — and builds each one from nothing. Nothing here assumes you've seen algebra before.
1. The variable — a box that holds a number
Plain words: is not a "thing" like an apple. It is a placeholder for a number. If I tell you "", then everywhere you see , you may write .
The picture: think of as a sealed box. You don't know what number is inside, but it is a number — so it obeys all the ordinary rules of numbers.
Why the topic needs it: the parent note wrote " and are like apples and oranges." But and aren't fruit — they are two different mystery boxes. We cannot merge boxes that might hold different numbers. That single fact is the whole reason "like terms" exists.
2. The coefficient — how many copies
Plain words: means "three copies of the box ", i.e.
The picture: three identical boxes lined up. If each box secretly holds the same number, three of them is three times that number.
Why the topic needs it: adding like terms is just counting copies. means "3 copies plus 5 copies = 8 copies" . The coefficient is the count that changes; the box is the thing being counted, and it never changes during addition.
3. The exponent — a box multiplied by itself
Careful — this is different from the coefficient!
- = three copies added:
- = three copies multiplied:
The picture: is the area of a square whose side is . is the volume of a cube. These are genuinely different sizes of thing.
Why the topic needs it: the parent warned against writing . Now you can see why it's wrong: you're counting boxes, and each box is "" (a cube). Two cubes plus five cubes is seven cubes — — not a bigger power. The exponent labels what kind of box; it is part of the "of what", so it never moves during addition.
Recall Why
and are unlike terms Because a square () and a cube () are different-shaped things. You can't merge their counts, just like you can't add a pile of coins to a pile of banknotes and call the total "coins".
4. Products of variables — and
Plain words: is " times ", a single combined box. is "". Two terms are the same kind of thing only if their variable parts match exactly, letter for letter and power for power.
| Term | Variable part | Same kind as ? |
|---|---|---|
| ✅ yes | ||
| ✅ yes | ||
| ❌ no (powers swapped) | ||
| ❌ no |
Why the topic needs it: Example 3 in the parent grouped , , together and kept separate. That grouping is only legal because the variable parts are identical. This idea is the gateway to Polynomials and later Factoring Algebraic Expressions.
5. The signs and , and parentheses
The key move — a in front of a bracket flips every sign inside:
The picture: think of the as a light switch applied to every term in the bracket, not just the first one.
Why this tool and not another? We use multiplication by (via the Distributive Property) rather than "just subtract the first term" because the parenthesis groups the whole expression as one object. Subtracting an object means reversing all of it — that's why the parent's Mistake 2 (forgetting to flip every sign) is so common. The Order of Operations tells us the bracket is one unit; distribution tells us how the reaches inside it.
How the foundations feed the topic
Read it as: variables + coefficients build terms; exponents and products decide each term's kind; matching kinds gives like terms; the sign rule handles subtraction — and together they power the parent topic.
Equipment checklist
Test yourself — cover the right side and answer out loud.