Foundations: Variables and Expressions
Build fluency with variables, algebraic expressions, order of operations, and translating between verbal descriptions and algebraic notation.
Learning Objectives
- 1Write and evaluate algebraic expressions using variables
- 2Translate verbal descriptions into algebraic expressions and equations
- 3Identify the parts of an expression: terms, coefficients, constants, and degree
- 4Apply the order of operations correctly to evaluate complex expressions
Before you learn what a variable is, watch mathematicians get genuinely excited about the simple idea of letters standing in for numbers. Makes variables feel like a secret code, not a chore.
Watch on YouTubeWelcome to Algebra
You have been working with numbers for years: adding, subtracting, multiplying, dividing. That is arithmetic. Algebra takes a fundamentally different step: instead of working with specific numbers, it works with relationships between numbers, even when those numbers are not yet known.
Here is the big idea that makes algebra so powerful: a variable is a placeholder for a number we do not know yet, or a number that can change. One letter can represent an entire range of possibilities.
Consider this: you work at a coffee shop earning $11.50 per hour. How much do you earn in a shift?
We could make a table:
- 3 hours: $34.50
- 4 hours: $46.00
- 5 hours: $57.50
- 6 hours: $69.00
Or we could write one expression that captures every row of the table at once:
The variable h represents any number of hours. Replace h with any value, and you immediately get the earnings. We have compressed an infinite table into a single, compact expression.
The Language of Expressions
An algebraic expression is a combination of numbers, variables, and operations. Let us break down a typical expression:
| Term | Role | Value |
|---|---|---|
| 5x² | First term | Coefficient is 5, variable is x, exponent is 2 |
| -3x | Second term | Coefficient is -3, variable is x, exponent is 1 |
| 7 | Third term | Constant (no variable attached) |
Key vocabulary:
- Term: A piece of the expression separated by addition or subtraction
- Coefficient: The number multiplied by a variable (5 and -3 above)
- Constant: A term with no variable (7 above)
- Degree: The highest exponent in the expression (degree 2 here, which makes it quadratic)
Think About
In the expression 8y - 12, what is the coefficient of y? What is the constant? If y = 3, what is the value of the entire expression?
Translating Between Words and Algebra
One of the most practical skills in algebra is moving between everyday language and algebraic notation. Here are the key phrases:
| Words | Operation | Example |
|---|---|---|
| "more than," "increased by," "sum of" | Addition (+) | "8 more than x" becomes x + 8 |
| "less than," "decreased by," "difference" | Subtraction (-) | "5 less than n" becomes n - 5 |
| "times," "product of," "multiplied by" | Multiplication | "7 times a number" becomes 7n |
| "divided by," "quotient of," "per" | Division | "x divided by 4" becomes x/4 |
| "twice," "double" | Multiply by 2 | "twice a number" becomes 2n |
| "squared" | Exponent of 2 | "a number squared" becomes n² |
Watch out for subtraction order! "5 less than n" means n - 5, NOT 5 - n. The word "than" reverses the order from how you read it. This is one of the most common mistakes in translating words to algebra.
Worked Example 1: Words to Algebra
A streaming service charges $14.99 per month plus a one-time sign-up fee of $5.00. Write an expression for the total cost after m months.
Step 1: Monthly cost for m months = 14.99 times m = 14.99m
Step 2: One-time fee = 5.00
Step 3: Total cost = 14.99m + 5.00
Worked Example 2: Algebra to Words
What does the expression 65t describe if t represents time in hours?
The coefficient 65 is multiplied by t (hours). This describes the distance traveled at 65 miles per hour. It is the classic distance formula d = rt, where the rate r = 65 mph.
Think About
A gym membership costs $30 per month plus a $50 enrollment fee. Write an expression for the total cost after m months. Then evaluate it for 6 months and 12 months. At what point does the enrollment fee become a small fraction of the total cost?
Order of Operations: The Grammar of Math
Without agreed-upon rules for the order of operations, the expression 3 + 2 x 5 could mean 25 or 13, depending on who reads it. Mathematics resolves this with PEMDAS:
- Parentheses (and other grouping symbols)
- Exponents
- Multiplication and Division (left to right)
- Addition and Subtraction (left to right)
Worked Example 3: Numerical
Evaluate: 4 + 3 x 2³
Step 1 (Exponents): 2³ = 8, giving 4 + 3 x 8
Step 2 (Multiplication): 3 x 8 = 24, giving 4 + 24
Step 3 (Addition): 4 + 24 = 28
Common mistake: doing 4 + 3 = 7 first, then 7 x 8 = 56. That ignores PEMDAS.
Worked Example 4: With Variables
Evaluate 3x² - 2x + 5 for x = 4.
Step 1 (Substitute): 3(4)² - 2(4) + 5
Step 2 (Exponent): 4² = 16, giving 3(16) - 2(4) + 5
Step 3 (Multiply): 3(16) = 48 and 2(4) = 8, giving 48 - 8 + 5
Step 4 (Left to right): 48 - 8 = 40, then 40 + 5 = 45
❓Concept Check
Evaluate 4x - 3y + x² for x = 5 and y = 2.
▸
Concept Check
Evaluate 4x - 3y + x² for x = 5 and y = 2.
Substitute: 4(5) - 3(2) + 5² = 20 - 6 + 25 = 39. Work through each piece: 4(5) = 20, 3(2) = 6, 5² = 25. Then combine left to right: 20 - 6 = 14, and 14 + 25 = 39.
Interactive graph — loading...
From Expressions to Equations
An expression has no equals sign: 3x + 7 is an expression.
An equation says two expressions are equal: 3x + 7 = 22 is an equation.
The equation makes a claim. It says that some value of x makes both sides balance. Finding that value is called solving the equation.
Think of an equation like a balance scale. Both pans must weigh the same. Whatever you do to one side, you must do to the other to keep it balanced.
To find x: subtract 7 from both sides to get 3x = 15, then divide both sides by 3 to get x = 5.
Check: 3(5) + 7 = 15 + 7 = 22. It balances.
We will develop all the formal solving techniques in Unit 3. For now, focus on this core idea: an equation is a balance, and algebra gives us legal moves that maintain that balance while isolating the unknown.
Worked Example 5: Writing an Equation from a Scenario
Your phone plan charges $45 per month. You also paid $199 for the phone itself. Write an equation that says your total spending equals $739.
Total spending = monthly cost times months + phone cost:
Solving: 45m = 540, so m = 12 months. After one year, you will have spent exactly $739.
Think About
A student earns $10 per hour tutoring. She also received a $25 gift card as a thank-you from one family. Write an equation that says her total earnings equal $175. What does solving that equation tell you in practical terms?
❓Concept Check
A rectangle has a perimeter of 36 inches. Its length is twice its width. Write an equation using the variable w for width, then find the dimensions.
▸
Concept Check
A rectangle has a perimeter of 36 inches. Its length is twice its width. Write an equation using the variable w for width, then find the dimensions.
Perimeter = 2(length) + 2(width). Since length = 2w: 2(2w) + 2w = 36. That simplifies to 4w + 2w = 36, then 6w = 36, so w = 6 inches. Length = 2(6) = 12 inches. Check: 2(12) + 2(6) = 24 + 12 = 36 inches.
Why This Matters Beyond the Classroom
The language of variables and expressions shows up everywhere:
- Science: The formula F = ma (force = mass times acceleration) uses variables to describe how objects move.
- Finance: Interest formulas use variables for principal, rate, and time — letting banks calculate returns for any deposit.
- Computer science: Every variable in a program is exactly what you learned here — a named placeholder for a value that can change.
- Sports analytics: Batting averages, quarterback ratings, and player efficiency ratings are all algebraic expressions applied to game statistics.
In Geometry, you will use algebraic expressions to write formulas for area, perimeter, and volume. The algebraic thinking you build in this unit is the foundation for every math course that follows.
Summary
- A variable is a symbol representing an unknown or changing quantity
- An algebraic expression combines variables, numbers, and operations — no equals sign
- Terms, coefficients, and constants are the structural parts of expressions
- Use PEMDAS (Parentheses, Exponents, Multiplication/Division, Addition/Subtraction) to evaluate correctly
- An equation sets two expressions equal and makes a claim about the value of a variable
- Translating between words and algebra is a fundamental skill that connects math to the real world