Introduction to Variables and Expressions
Take your first steps into the world of algebra
Many people found themselves enjoying math right up until they encountered algebra. So, if letters mixed in with numbers seems scary to you, know that you are not alone. But here is the thing you need to realize: you already think in variables every single day. When you wonder “how much will this cost?” or “how long until we get there?” - that unknown value you are thinking about is a variable. We are just going to give it a name and learn to work with it on paper.
Core Concepts
What is a Variable?
A variable is simply a letter (or symbol) that stands in for a number we do not know yet, or a number that can change. That is it. Nothing magical, nothing intimidating - just a placeholder.
Think about this: You check the price of a concert ticket online, and it says “$45 + fees.” Those “fees” are unknown to you right now. If we called the fees $x$, then the total cost would be $45 + x$. That is algebra. You have been reading algebraic expressions your whole life without calling them that.
Common letters used as variables: $x$, $y$, $n$, $t$, $a$, $b$
Why do we use variables? Because they let us:
- Talk about unknown values before we figure them out
- Write rules that work for many different numbers at once
- Describe relationships between quantities
What is an Algebraic Expression?
An algebraic expression is a combination of numbers, variables, and operations (like addition, subtraction, multiplication, and division). It is like a recipe that tells you what to do with numbers - even when you do not know what all the numbers are yet.
Here are some examples of algebraic expressions:
- $3x + 5$ (three times some number, plus five)
- $2y - 7$ (two times some number, minus seven)
- $\frac{n}{4} + 10$ (some number divided by four, plus ten)
- $5(a + 2)$ (five times the quantity of some number plus two)
Notice: An expression does not have an equals sign. Once you add an equals sign, it becomes an equation - but that is a topic for another chapter.
Writing Algebraic Expressions from Words
One of the most useful skills in algebra is translating everyday language into mathematical expressions. Here is a guide to common phrases:
| Words | Operation | Example in Words | Expression |
|---|---|---|---|
| sum, plus, more than, increased by, total | Addition | five more than a number | $x + 5$ |
| difference, minus, less than, decreased by | Subtraction | a number decreased by 3 | $x - 3$ |
| product, times, of, multiplied by | Multiplication | twice a number | $2x$ |
| quotient, divided by, per, ratio | Division | a number divided by 7 | $\frac{x}{7}$ |
Watch out for “less than”! The phrase “5 less than a number” means $x - 5$, not $5 - x$. You start with the number and take away 5.
Evaluating Expressions
Evaluating an expression means finding its value when you replace the variable with a specific number. Think of it as filling in the blank.
If $x = 4$, then $3x + 2 = 3(4) + 2 = 12 + 2 = 14$.
You are not solving for $x$ here - someone is telling you what $x$ equals, and you are calculating the result.
Notation and Terminology
| Term | Meaning | Example |
|---|---|---|
| Variable | A letter representing an unknown or changing value | In $3x + 5$, the variable is $x$ |
| Expression | A combination of numbers, variables, and operations | $4y - 7$ |
| Term | A single number, variable, or number times a variable | In $3x + 5$, the terms are $3x$ and $5$ |
| Coefficient | The number multiplied by a variable | In $7n$, the coefficient is 7 |
| Constant | A term without a variable (just a number) | In $2x + 9$, the constant is 9 |
| Like terms | Terms with the same variable(s) to the same power | $3x$ and $5x$ are like terms; $3x$ and $3y$ are not |
| Simplify | Combine like terms to write an expression more simply | $4x + 3x = 7x$ |
A note about multiplication notation: In algebra, we usually do not write the multiplication sign because it looks too much like the variable $x$. Instead of $3 \times x$, we write $3x$. Instead of $a \times b$, we write $ab$. When multiplying a number by itself, we use exponents: $x \times x = x^2$.
Examples
Identify the terms, coefficients, and constants in the expression $5x + 3y - 8$.
Solution:
Let us break this down piece by piece:
Terms: The parts separated by + and - signs
- $5x$ (first term)
- $3y$ (second term)
- $8$ (third term - remember, we are subtracting it)
Coefficients: The numbers in front of variables
- The coefficient of $x$ is 5
- The coefficient of $y$ is 3
Constant: The term without a variable
- The constant is $-8$ (keep the negative sign with it!)
Think of it like a recipe: the coefficients tell you how much of each “ingredient” (variable) you have, and the constant is what you add regardless of those ingredients.
Write an algebraic expression for: “Seven more than three times a number.”
Solution:
Let us decode this phrase by phrase:
- “a number” - this is our unknown, so let us call it $n$
- “three times a number” - that is $3n$
- “seven more than” - this means we add 7
Putting it together: $3n + 7$
Check your understanding: This expression means: take some number, multiply it by 3, then add 7. If the number happened to be 10, you would get $3(10) + 7 = 37$.
Evaluate $4x^2 - 3x + 5$ when $x = 2$.
Solution:
This is like following a recipe where someone finally told you what $x$ is. Everywhere you see $x$, plug in 2:
$$4x^2 - 3x + 5$$
$$= 4(2)^2 - 3(2) + 5$$
Now follow the order of operations (exponents first, then multiplication, then addition/subtraction):
$$= 4(4) - 3(2) + 5$$
$$= 16 - 6 + 5$$
$$= 15$$
Pro tip: Use parentheses when substituting to avoid mistakes. Writing $4(2)^2$ instead of $4 \cdot 2^2$ makes it clearer that you square the 2 first.
Simplify: $6x + 4 - 2x + 9$
Solution:
First, identify like terms:
- $6x$ and $-2x$ are like terms (both have $x$)
- $4$ and $9$ are like terms (both are constants)
Now combine each group:
- $6x - 2x = 4x$ (think: 6 apples minus 2 apples = 4 apples)
- $4 + 9 = 13$
Final answer: $4x + 13$
Why this works: You can only combine terms that have the same variable. Just like you cannot add 3 apples + 2 oranges and get “5 apple-oranges,” you cannot combine $3x + 2y$ into a single term.
Simplify: $3(2x + 5) - 2(x - 4)$
Solution:
The distributive property says that when you multiply a number by a sum (or difference), you can “distribute” that multiplication to each term inside the parentheses:
$$a(b + c) = ab + ac$$
Step 1: Apply the distributive property to each set of parentheses.
For $3(2x + 5)$: multiply both $2x$ and $5$ by 3 $$3(2x + 5) = 3 \cdot 2x + 3 \cdot 5 = 6x + 15$$
For $-2(x - 4)$: multiply both $x$ and $-4$ by $-2$ $$-2(x - 4) = -2 \cdot x + (-2) \cdot (-4) = -2x + 8$$
Be careful with signs! A negative times a negative gives a positive, so $-2 \times (-4) = +8$.
Step 2: Combine everything. $$6x + 15 - 2x + 8$$
Step 3: Combine like terms. $$6x - 2x + 15 + 8 = 4x + 23$$
Final answer: $4x + 23$
Key Properties and Rules
The Distributive Property
This is one of the most important tools in algebra. It lets you remove parentheses:
$$a(b + c) = ab + ac$$ $$a(b - c) = ab - ac$$
It works in reverse too - you can “factor out” a common number: $$6x + 9 = 3(2x + 3)$$
Combining Like Terms
Terms are “like” if they have exactly the same variable(s) with the same exponent(s):
- $5x$ and $-3x$ are like terms (combine to get $2x$)
- $4x^2$ and $7x^2$ are like terms (combine to get $11x^2$)
- $2x$ and $2x^2$ are not like terms (different exponents)
- $3xy$ and $5xy$ are like terms (combine to get $8xy$)
Order of Operations with Variables
When evaluating expressions, always follow PEMDAS/BODMAS:
- Parentheses (Brackets)
- Exponents (Orders)
- Multiplication and Division (left to right)
- Addition and Subtraction (left to right)
Simplifying Expressions
To fully simplify an expression:
- Apply the distributive property to remove all parentheses
- Combine all like terms
- Write the final expression in standard form (usually terms with variables first, in descending order of exponents, then constants)
Real-World Applications
Shopping and Budgeting
You have $50 to spend. If shirts cost $x each, the expression $50 - 3x$ tells you how much money you have left after buying 3 shirts. If shirts cost $12 each, you would have $50 - 3(12) = 50 - 36 = $14$ left.
Cell Phone Plans
A phone plan charges $30 per month plus $0.05 per text over your limit. If $t$ represents texts over your limit, your bill is $30 + 0.05t$. Go 100 texts over? Your bill is $30 + 0.05(100) = 30 + 5 = $35$.
Cooking and Recipes
A recipe for 4 servings calls for 2 cups of rice. If you need $n$ servings, you need $\frac{2n}{4} = \frac{n}{2}$ cups of rice. For 10 servings, that is $\frac{10}{2} = 5$ cups.
Sports Statistics
A basketball player scores an average of $p$ points per game. After $g$ games, total points = $pg$. If they average 22 points and have played 15 games, total points = $22 \times 15 = 330$.
Distance and Travel
If you are driving at $s$ miles per hour for $t$ hours, the distance traveled is $st$ miles. At 65 mph for 3 hours, you cover $65 \times 3 = 195$ miles.
Self-Test Problems
Problem 1: Write an algebraic expression for “twelve less than four times a number.”
Show Answer
Let the number be $n$.
- “Four times a number” is $4n$
- “Twelve less than” means subtract 12 from that
The expression is $4n - 12$.
Remember: “Less than” means you subtract from what comes after it!
Problem 2: Evaluate $2x^2 + 3x - 7$ when $x = 3$.
Show Answer
Substitute $x = 3$:
$$2(3)^2 + 3(3) - 7$$ $$= 2(9) + 9 - 7$$ $$= 18 + 9 - 7$$ $$= 20$$
Problem 3: Simplify by combining like terms: $8a - 3b + 2a + 5b - 4$
Show Answer
Group like terms:
- Terms with $a$: $8a + 2a = 10a$
- Terms with $b$: $-3b + 5b = 2b$
- Constants: $-4$
Final answer: $10a + 2b - 4$
Problem 4: Use the distributive property to simplify: $5(x + 3) + 2(x - 1)$
Show Answer
Apply the distributive property: $$5(x + 3) = 5x + 15$$ $$2(x - 1) = 2x - 2$$
Combine: $$5x + 15 + 2x - 2$$
Combine like terms: $$7x + 13$$
Problem 5: A gym charges a $50 registration fee plus $m per month. Write an expression for the total cost after 6 months, then find the cost if the monthly fee is $25.
Show Answer
The expression for total cost after 6 months is: $$50 + 6m$$
If $m = 25$: $$50 + 6(25) = 50 + 150 = $200$$
Summary
- A variable is a letter that represents an unknown or changing number - it is just a placeholder until you know the actual value.
- An algebraic expression combines numbers, variables, and operations without an equals sign.
- Terms are the parts of an expression separated by + and - signs. Coefficients are the numbers multiplied by variables. Constants are terms without variables.
- To evaluate an expression, substitute the given value for the variable and calculate.
- Like terms have the same variable(s) to the same power. You can combine like terms by adding or subtracting their coefficients.
- The distributive property ($a(b + c) = ab + ac$) lets you remove parentheses by multiplying each term inside by the number outside.
- To simplify an expression: distribute to remove parentheses, then combine all like terms.
- Variables and expressions are everywhere in real life - prices, distances, recipes, statistics - giving names to unknown values so we can work with them before we know their exact amounts.