Variables and Constants

So a variable is just a box that holds a number. That's literally it. We use letters like x, y, or n because it looks more mathy, but it's the same idea. The number inside the box might be something we don't know yet, or something that can change. Either way, the letter is just standing in for it.

A constant, on the other hand, is a plain number that stays fixed. In 3x + 7, the variable is x and the constant is 7. It doesn't move, it doesn't change. It's just 7. And here's why variables are actually useful: because x can be anything, a single algebraic formula can solve every version of a problem at once. P = 4s works for every square ever made, no matter the size. Plug in the side length, done.

Expression: 5n − 3
Variable: n (unknown or changing value)
Constant: 3 (fixed)
Coefficient: 5 (the number multiplying n)

Reading Algebraic Notation

Algebra has a bunch of shorthand that nobody ever really explains out loud. And honestly, once you know the rules, it stops being confusing. Here's what you need to know:

  • Numbers touching a variable mean multiplication - writing 4x means 4 times x. You don't need a multiplication sign. The two things being next to each other is enough.
  • The number in front of a variable is its coefficient - in 9y, the coefficient is 9. It's telling you "nine of these y things."
  • Exponents only apply to what's directly below them - in 3x², that 2 belongs to x only. So it's 3 × x × x, not (3x)².
  • A lone variable has a hidden 1 in front - writing x is the same as 1x. The 1 is just invisible.
  • Division can be written as a fraction - x/4 and ¼x mean the same thing. Pick whichever looks neater.
Term: −6ab²
Coefficient: −6
Variables: a and b
Exponent on b: 2 (so b is multiplied by itself)
Exponent on a: 1 (implied)

Combining Like Terms

Here's a concept that trips people up at first. Like terms are terms that have exactly the same variable part. Same letters, same exponents. If they match on that, they're "like" and you can add or subtract them. If they don't match, they stay separate.

Think of it like this: 3 apples + 5 apples = 8 apples. But 3 apples + 5 oranges? You can't mash those together. You still have 3 apples and 5 oranges sitting there separately. Same thing in algebra. x-terms and y-terms don't mix.

Simplify: 7x² + 3x − 2x² + 5 − x

Group like terms:
x² terms: 7x² − 2x² = 5x²
x terms: 3x − x = 2x
Constants: 5

Result: 5x² + 2x + 5

And yeah, x² and x are not like terms. One's squared, one isn't. Different exponents means they stay separate, even though they both have x in them.

The Distributive Property

The distributive property is something you'll use constantly. The idea is simple: if there's a number outside parentheses, it multiplies everything inside. Not just the first thing. Everything.

a(b + c) = ab + ac

Example 1: 4(x + 3) = 4x + 12
Example 2: −2(3y − 5) = −6y + 10
Example 3: x(x + 7) = x² + 7x

The place people go wrong is negative signs. If you have −(x − 4), that minus sign out front multiplies every term inside. So you get −x + 4, not −x − 4. The double negative on that 4 flips it positive. Don't skip over that.

This also runs in reverse, which is called factoring. If every term in an expression shares a common factor, you can pull it out front: 6x + 9 = 3(2x + 3). It's the same expression, just written differently.

Substitution

Substitution is exactly what it sounds like: you swap the variable out for an actual number, then do the arithmetic. It's how you check your answers, and it's how formulas get applied to real problems.

Evaluate 4x² − 3x + 1 when x = 2

Step 1 - Replace x with 2:
4(2)² − 3(2) + 1

Step 2 - Apply exponent first:
4(4) − 3(2) + 1

Step 3 - Multiply:
16 − 6 + 1

Step 4 - Add and subtract left to right:
Answer = 11

One thing: don't skip the order of operations. PEMDAS, BODMAS, whatever your school calls it. Brackets first, then exponents, then multiply and divide, then add and subtract. Honestly, most substitution mistakes come from people just rushing through in the wrong order.