Foundations: Variables and Equations
Build fluency with variables, algebraic expressions, and the structure of equations as the language of algebra.
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
- 4Apply the order of operations correctly to evaluate expressions
What Is Algebra, Really?
You've been doing arithmetic for years: 3 + 4 = 7, 12 × 5 = 60. Arithmetic works on specific numbers. Algebra goes further — it works on relationships between numbers, even when the numbers aren't known yet.
Here's the big idea: a variable is a placeholder for a number we don't know yet (or a number that can change). That single idea unlocks an enormous amount of mathematical power.
Consider this problem: You earn $12 per hour babysitting. How much do you earn in a given number of hours?
We could list it out:
- 1 hour → $12
- 2 hours → $24
- 3 hours → $36
- 10 hours → $120
But algebra lets us capture all of these at once with one expression:
The variable h represents any number of hours. Plug in any value of h and you immediately get the earnings. We've compressed an infinite table into a single expression.
Key Vocabulary: Parts of an Expression
An algebraic expression is a combination of numbers, variables, and operations. Let's break down a typical expression:
- Terms: The parts separated by addition or subtraction — here, 5x², −3x, and 7
- Coefficient: The number multiplied by a variable — 5 is the coefficient of x², −3 is the coefficient of x
- Constant: A term with no variable — here, 7
- Variable: The letter x (it can take different values)
- Degree: The highest exponent — here, degree 2 (called a quadratic)
Think About
In the expression 4y - 9, what is the coefficient of y? What is the constant? If y = 5, what is the value of the whole expression?
Translating Between Words and Algebra
One of the most important skills in algebra is moving between verbal descriptions and algebraic notation. Here are the key phrases to know:
| Words | Operation | Example |
|---|---|---|
| "more than," "increased by," "sum of" | + | "8 more than x" → x + 8 |
| "less than," "decreased by," "difference" | − | "5 less than n" → n − 5 |
| "times," "product of," "multiplied by" | × | "7 times a number" → 7n |
| "divided by," "quotient of," "per" | ÷ | "x divided by 4" → x/4 |
Watch out for subtraction! "5 less than n" means n − 5, NOT 5 − n. The word "than" reverses the order.
Worked Example: Words to Algebra
The school store sells pencils for $0.25 each and notebooks for $1.50 each. Write an expression for the total cost of buying p pencils and n notebooks.
Step 1: Cost of pencils = 0.25 × p = 0.25p
Step 2: Cost of notebooks = 1.50 × n = 1.50n
Step 3: Total cost = 0.25p + 1.50n
Worked Example: Algebra to Words
What does the expression 60t describe if t represents time in hours?
The coefficient 60 is multiplied by t (hours). This describes the distance traveled at a speed of 60 miles per hour — the classic distance formula d = rt, where r = 60 mph.
Think About
A runner burns approximately 100 calories per mile. Write an expression for total calories burned after running m miles. Then evaluate the expression for 5 miles, 10 miles, and a half marathon (13.1 miles).
Order of Operations: The Grammar of Algebra
Order of operations — often remembered as PEMDAS (Parentheses, Exponents, Multiplication/Division, Addition/Subtraction) — is the agreed-upon set of rules that tells us which operations to perform first. Without these rules, the same expression could mean different things to different people.
Evaluate: 3 + 2 × 5²
Step 1 (Exponents): 5² = 25 → 3 + 2 × 25
Step 2 (Multiplication): 2 × 25 = 50 → 3 + 50
Step 3 (Addition): 3 + 50 = 53
Common mistake: doing left-to-right without respecting order. 3 + 2 = 5, then 5 × 25 = 125 → wrong!
Worked Example with Variables
Evaluate 2x² + 3x − 1 for x = 4.
Step 1 (Exponent first): x² = 4² = 16 → 2(16) + 3(4) − 1
Step 2 (Multiplication): 2(16) = 32 and 3(4) = 12 → 32 + 12 − 1
Step 3 (Left to right): 32 + 12 = 44, then 44 − 1 = 43
❓Concept Check
Evaluate 5x - 2y + x² for x = 3 and y = 4.
▸
Concept Check
Evaluate 5x - 2y + x² for x = 3 and y = 4.
Substitute: 5(3) - 2(4) + 3² = 15 - 8 + 9 = 16
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 equal. Our job in algebra will often be to find that value. This is called solving the equation.
Think of an equation like a scale. Both pans must balance. Whatever you do to one side, you must do to the other.
To find x: subtract 7 from both sides → 3x = 15 → divide both sides by 3 → x = 5.
Check: 3(5) + 7 = 15 + 7 = 22. ✓
We'll develop all of these solving techniques in Unit 3. For now, focus on the idea: an equation is a balance, and algebra is the set of legal moves that keep the balance while isolating the unknown.
Think About
A student earns $9 per hour at a part-time job. She also received a one-time $20 sign-on bonus. Write an equation that says her total earnings equal $200. What does solving that equation tell you?
Khan Academy Resource: For additional practice with variables and expressions, try the Introduction to Algebra section on Khan Academy Algebra 1. The "Writing algebraic expressions" exercises are especially helpful.
Why This Matters Beyond Algebra
The language of variables and expressions appears everywhere:
- Science: The formula d = rt (distance = rate × time) lets physicists describe motion for any speed and time.
- Finance: Compound interest formulas use variables for principal, rate, and time.
- Computer science: Every variable in a program is exactly what you learned here — a named placeholder for a value.
- Statistics: Data scientists write expressions to transform and analyze datasets.
In Geometry next year, you'll use algebraic expressions to write formulas for area, perimeter, and volume — and you'll see that the algebraic thinking you build here is the foundation for everything that follows.
Summary
- A variable is a symbol representing an unknown or changing quantity
- An algebraic expression combines variables, numbers, and operations
- Terms, coefficients, and constants are the structural parts of expressions
- Use PEMDAS to evaluate expressions correctly
- An equation sets two expressions equal and makes a claim about the value of a variable
