Table of Contents
In pre-algebra, variables and expressions are the basic “language pieces” of algebra. Numbers, operations, and symbols are combined to describe quantities and relationships without always using specific, fixed numbers.
This chapter will introduce what variables are, how they differ from simple numbers, what algebraic expressions look like, and how they are built.
Variables: Unknown and Changing Numbers
A variable is a symbol (usually a letter) that stands for a number. The actual number may be:
- unknown,
- allowed to change,
- or not yet chosen.
Common variable symbols: $x, y, z, a, b, c$.
Examples:
- $x$ could be the number of apples you buy.
- $t$ could be the time in seconds.
- $n$ could be any whole number.
A variable is not a “mystery forever”; it is simply a placeholder. Once a value is chosen or given, the variable then has that value.
For example, if $x$ represents the number of students in a room and there are 24 students, then $x = 24$ in that situation.
Constants and Coefficients
A constant is a fixed number, such as $2$, $-5$, or $10.75$. It does not change in a given problem.
When constants are multiplied by variables, the constant part has a special name: a coefficient.
- In $3x$, $3$ is the coefficient and $x$ is the variable.
- In $-7y$, $-7$ is the coefficient and $y$ is the variable.
- In $0.5t$, $0.5$ is the coefficient and $t$ is the variable.
If a variable appears without a visible coefficient, its coefficient is $1$:
- $x$ really means $1x$.
- $-y$ really means $-1y$.
Algebraic Expressions: No Equals Sign
An algebraic expression is a combination of:
- numbers,
- variables, and
- operation symbols ($+, -, \times, \div$, exponents, etc.)
but without an equals sign.
Examples of algebraic expressions:
- $x + 5$
- $3y - 2$
- $4a + 7b$
- $2n^2 - 3n + 1$
- $\dfrac{m}{4} + 6$
Non-examples (these are equations, not expressions, because they have equals signs):
- $x + 5 = 12$
- $3y - 2 = 10$
In this chapter we focus on expressions themselves, not solving equations. Equations are treated in the “Linear Equations” chapter.
Terms and Factors in Expressions
Expressions are built out of smaller pieces called terms.
A term in an expression is a single piece separated from others by $+$ or $-$ signs (but those plus or minus signs are not part of the term itself).
Examples:
- In $3x + 5$, the terms are $3x$ and $5$.
- In $2a - 7$, the terms are $2a$ and $7$ (the $-$ belongs to the operation, not to the 7 when we list the terms).
- In $4m^2 - 3m + 8$, the terms are $4m^2$, $3m$, and $8$.
Inside a term, multiplication is formed by factors.
A factor is a part of a multiplication.
- In $3x$, the factors are $3$ and $x$.
- In $5ab$, the factors are $5$, $a$, and $b$.
- In $-2x^2y$, the factors are $-2$, $x^2$, and $y$.
Notice that exponents stay attached to their base as one factor ($x^2$ is one factor, not two).
Implicit Multiplication (Hidden Multiplication Sign)
In algebra, multiplication is often written without the $\times$ symbol when variables are involved.
- $3 \times x$ is usually written as $3x$.
- $a \times b$ is usually written as $ab$.
- $2 \times x \times y$ is often written as $2xy$.
You almost never see $3 \times x$ written with the $\times$ sign in algebra, because it can be confused with the letter $x$.
Multiplication can still be shown with parentheses:
- $3(x + 2)$ means $3$ multiplied by $(x + 2)$.
- $(a + 1)(b - 4)$ means $(a + 1)$ multiplied by $(b - 4)$.
Common Forms of Simple Expressions
Algebraic expressions often appear in a few common patterns.
Sum and Difference Forms
A number added to or subtracted from a variable or from a multiple of a variable:
- $x + 3$ (“a number $x$ increased by 3”)
- $5 - y$ (“5 decreased by a number $y$”)
- $4n - 7$ (“4 times $n$ minus 7”)
Product Forms
A constant times a variable or several variables:
- $7x$ (“7 times $x$”)
- $-2ab$ (“negative 2 times $a$ times $b$”)
- $\dfrac{1}{3}t$ (“one third of $t$”)
Power Forms
A variable raised to a power (often 2 or 3 in early algebra):
- $x^2$ (“$x$ squared”)
- $y^3$ (“$y$ cubed”)
- $5x^2$ (“5 times $x$ squared”)
Powers tell you repeated multiplication of the same factor, but the detailed laws of exponents belong to another chapter.
Fractional Forms
Variables and constants can appear in numerators and denominators:
- $\dfrac{x}{2}$
- $\dfrac{3}{y}$
- $\dfrac{a + 1}{b - 2}$
Manipulating these more deeply involves fraction skills, which are covered in the fractions chapter.
Verbal Descriptions to Expressions
A big skill in pre-algebra is turning word phrases into algebraic expressions. Here we focus on common wording patterns; word problems in full will be treated in later chapters.
Key phrases and their typical meanings:
- “sum of” $\to$ use $+$
“the sum of $x$ and $” $\to$ $x + 5$. - “difference of” $\to$ use $-$
“the difference of $ and $y$” $\to$ - y$. - “product of” $\to$ use $\times$ (usually written without $\times$)
“the product of $ and $n$” $\to$ n$. - “quotient of” $\to$ use division
“the quotient of $x$ and $” $\to$ $\dfrac{x}{4}$.
Other common phrases:
- “increased by” $\to$ add something
“a number $x$ increased by 9” $\to$ $x + 9$. - “decreased by” $\to$ subtract something
“a number $n$ decreased by 3” $\to$ $n - 3$. - “more than” often means add, but be careful with order:
“5 more than a number $k$” $\to$ $k + 5$. - “less than” often reverses the order:
“3 less than a number $p$” $\to$ $p - 3$.
It is important to pay attention to the order of the words. For example:
- “$7$ less than $x$” $\to$ $x - 7$, not $7 - x$.
- “$4$ more than $y$” $\to$ $y + 4$, not $4 + y$ (though addition is commutative, writing it as $y + 4$ keeps the meaning clearer).
Using Parentheses in Expressions
Parentheses group parts of an expression so they are treated as a single unit.
- $3(x + 2)$ means multiply $3$ by the whole sum $x + 2$.
- $2(a - 5)$ means $2$ times the entire difference $a - 5$.
- $(x + 1)(x - 1)$ means the product of two binomials.
Parentheses change how operations are interpreted. For example, $3x + 2$ and $3(x + 2)$ are different expressions:
- $3x + 2$ means “three times $x$, then add 2.”
- $3(x + 2)$ means “first add 2 to $x$, then multiply the result by 3.”
The detailed rules about the order of operations and how parentheses affect it are handled in the arithmetic part of the course; here, you should recognize that parentheses can combine several pieces of an expression into one “chunk.”
Like Terms and Unlike Terms (Introduction)
Expressions often have several terms involving the same variable raised to the same power. These are called like terms.
- $3x$ and $5x$ are like terms (both are “something times $x$”).
- $2y^2$ and $-7y^2$ are like terms (both are “something times $y^2$”).
Examples with unlike terms:
- In $3x + 2x^2$, $3x$ and $2x^2$ are not like terms, because $x$ and $x^2$ are different powers.
- In $4a + 3b$, the terms $4a$ and $3b$ are unlike terms, because $a$ and $b$ are different variables.
Recognizing like and unlike terms is a stepping stone to simplifying expressions, which is continued in later work.
Expressions vs. Equations vs. Inequalities
It is helpful to clearly distinguish three common objects:
- Expression: has no equals or inequality sign.
Example: x + 5$. - Equation: has an equals sign and states that two expressions are equal.
Example: x + 5 = 20$. - Inequality: uses symbols like $<, >, \le, \ge$ to compare expressions.
Example: y - 1 > 7$.
In this chapter, we stay with expressions themselves. Solving equations and inequalities is handled in other chapters.
Describing Real Situations with Variables and Expressions
Variables allow us to describe changing or unknown quantities in real-life situations.
Examples:
- If each notebook costs \$3 and you buy $n$ notebooks, the total cost (in dollars) can be written as the expression $3n$.
- If a car travels at a constant speed of $60$ km/h for $t$ hours, the distance traveled is $60t$ km.
- If you are $4$ years older than your brother and his age is $b$, your age can be written as $b + 4$.
In each case:
- The letters ($n, t, b$) stand for quantities that can change,
- The expressions ($3n$, $60t$, $b + 4$) represent how other quantities depend on them.
This idea of using variables and expressions to model relationships is at the heart of algebra and will be used throughout pre-algebra and beyond.