Variables and Expressions: Letters That Stand for Numbers

A variable is not an unknown. It's a slot.

Here's what confuses people: they think x means "mystery number" — something hidden that you need to discover. But x is just a placeholder. It's the blank in a form, waiting to be filled in.

The unlock: a variable is a slot that can hold any number. Sometimes you're trying to find which number fits (solving equations). Sometimes you're describing what happens for all numbers (writing formulas). Either way, the variable is just marking where a number goes.

Once you see variables as slots rather than mysteries, algebra becomes fill-in-the-blank with rules.

The Slot Mental Model

Think of a variable like a blank in a sentence:

"I bought ___ apples."

The blank isn't mysterious. It's waiting for a number. You might fill it with 3, or 7, or 0. The sentence structure stays the same; only the number changes.

Algebraic expressions work the same way:

"Three times ___ plus five" becomes 3x + 5.

The x is the blank. Plug in 2, you get 3(2) + 5 = 11. Plug in 10, you get 3(10) + 5 = 35. The expression is a template; the variable is where numbers go.

Why Letters?

Why use x instead of a blank or a box?

Because letters can be manipulated. You can write x + x = 2x, which tells you that doubling the slot doubles the result. You can write xy to mean "the first slot times the second slot."

Letters follow the same rules as numbers. You can add them, multiply them, factor them. This lets you do arithmetic with slots — figuring out relationships before you know which specific numbers are involved.

The choice of which letter doesn't matter mathematically. x, y, n, t — they're all just labels. Convention assigns certain letters to certain contexts (t for time, n for integers), but the math works regardless.

Expressions: Templates for Calculation

An expression is a mathematical phrase — a recipe for computing something once you know the variables' values.

Examples:

• 3x + 5 (take x, triple it, add 5)
• x² - 4 (take x, square it, subtract 4)
• 2(a + b) (take a and b, add them, double the result)

Expressions don't have equals signs. They're not statements that are true or false. They're instructions: "given this input, here's the output."

Evaluating Expressions

To evaluate an expression, plug in numbers for the variables and calculate.

Expression: 3x + 5 If x = 2: 3(2) + 5 = 6 + 5 = 11 If x = -1: 3(-1) + 5 = -3 + 5 = 2 If x = 0: 3(0) + 5 = 0 + 5 = 5

Same expression, different inputs, different outputs. The expression is the machine; evaluation is running it.

Coefficients and Constants

In an expression like 3x + 5:

Coefficient: The number multiplied by a variable. Here, 3 is the coefficient of x. It tells you "how many x's."

Constant: A number standing alone, not attached to a variable. Here, 5 is a constant. It doesn't change when x changes.

In 7x² - 4x + 9:

• 7 is the coefficient of x²
• -4 is the coefficient of x
• 9 is the constant

Coefficients scale the variable. Constants shift the result.

Terms

An expression is made of terms — pieces separated by addition or subtraction.

In 3x + 5, the terms are 3x and 5.

In 4x² - 2x + 7, the terms are 4x², -2x, and 7.

Terms are the building blocks. Each term is either a constant or a variable with a coefficient.

Like Terms

Like terms have the same variable raised to the same power.

• 3x and 5x are like terms (both have x¹)
• 2x² and -7x² are like terms (both have x²)
• 3x and 3x² are NOT like terms (different powers)
• 4x and 4y are NOT like terms (different variables)

You can only combine like terms:

3x + 5x = 8x ✓ 2x² + 3x² = 5x² ✓ 3x + 2x² = 3x + 2x² (can't simplify — unlike terms)

This is like adding apples to apples. Three apples plus five apples is eight apples. But three apples plus five oranges is just... three apples and five oranges.

Simplifying Expressions

To simplify an expression, combine like terms and apply arithmetic.

Example: 4x + 3 + 2x - 1

Identify like terms:

• 4x and 2x (both x terms)
• 3 and -1 (both constants)

Combine:

• 4x + 2x = 6x
• 3 + (-1) = 2

Simplified: 6x + 2

The expression is equivalent — it gives the same output for any input. It's just written more compactly.

The Distributive Property

The distributive property connects multiplication and addition:

a(b + c) = ab + ac

Multiplying a sum is the same as summing the products.

Example: 3(x + 4) = 3x + 12

You "distribute" the 3 to each term inside the parentheses.

This works in reverse too — factoring:

6x + 9 = 3(2x + 3)

The 3 was distributed into 6x and 9. You can factor it back out.

Order of Operations

Expressions are evaluated in a specific order (PEMDAS):

1. Parentheses
2. Exponents
3. Multiplication and Division (left to right)
4. Addition and Subtraction (left to right)

Example: 3 + 4 × 2²

• First, exponent: 2² = 4
• Then multiplication: 4 × 4 = 16
• Then addition: 3 + 16 = 19

Not 3 + 4 = 7, then 7 × 4 = 28. Order matters.

Substitution

Substitution means replacing a variable with an expression.

If y = 2x + 1, and you have an expression 3y - 5, you can substitute:

3y - 5 = 3(2x + 1) - 5 = 6x + 3 - 5 = 6x - 2

You've rewritten the expression in terms of x instead of y.

This is how you connect expressions — plug one into another.

Expressions Describe Relationships

Expressions aren't just for solving equations. They describe how quantities relate.

"A rectangle's area is length times width" → A = lw

"Distance is speed times time" → d = rt

"The cost of n items at $5 each, plus $3 shipping" → 5n + 3

Before you solve anything, you need an expression that captures the relationship. The expression is the mathematical model of the situation.

Multiple Variables

Expressions can have multiple variables:

xy (the product of x and y) a² + b² (sum of squares of a and b) 2l + 2w (perimeter of a rectangle with length l and width w)

Each variable is an independent slot. Evaluate by plugging in values for all of them:

If a = 3 and b = 4, then a² + b² = 9 + 16 = 25.

The Core Insight

Variables are slots. Expressions are templates.

A variable marks where a number will go. An expression describes what to do with that number once you have it.

When you write 3x + 5, you're not solving anything yet. You're describing a process: "triple the input and add 5." The expression captures the relationship. Solving comes later.

This is why algebra is powerful. You can reason about relationships before you know the specific numbers. You can prove that certain things are always true, for any values in the slots.

The letters aren't hiding anything. They're holding space.

Part 2 of the Algebra Fundamentals series.

Previous: What Is Algebra? The Art of Finding What You Do Not Know Next: Solving Linear Equations: Isolating the Unknown