Chapter 5: Variables, Expressions & Properties
Algebra begins the moment we replace specific numbers with letters. In this chapter you will learn what variables and expressions mean, how to evaluate them for given values, and which fundamental properties of arithmetic make algebraic manipulation possible. By the end you will be comfortable simplifying expressions and translating real-world sentences into algebraic language — skills that underpin every mathematics course that follows.
5.1 Variables and Constants
Definition — Variable
A variable is a letter or symbol that represents an unknown or changing quantity. Common choices are $x$, $y$, $n$, and $t$, though any letter may be used.
Definition — Constant
A constant is a fixed number whose value does not change. Examples: $5$, $-3$, $\tfrac{2}{7}$, $\pi$.
A coefficient is the numerical factor multiplied by a variable. In $7x$, the coefficient is $7$. In $-3y^2$, the coefficient is $-3$. When no number appears in front of a variable (as in just $x$), the coefficient is understood to be $1$.
Definition — Algebraic Expression
An algebraic expression is a combination of variables, constants, and arithmetic operations ($+$, $-$, $\times$, $\div$, exponents). Examples:
- $3x + 5$ — one variable, one constant
- $2a^2 - 4b + 1$ — two variables, degree 2
- $\dfrac{n+3}{2}$ — fraction expression
An expression does not contain an equals sign. Once you add an $=$, you have an equation.
5.2 Evaluating Expressions
To evaluate an expression means to substitute specific numbers for each variable and then simplify using the correct order of operations (PEMDAS).
Example 1 — Single Variable
Evaluate $4x - 7$ when $x = 3$.
Solution:
Substitute $x = 3$:
$$4(3) - 7 = 12 - 7 = 5$$Example 2 — Two Variables
Evaluate $2a^2 + 3b$ when $a = -2$ and $b = 5$.
Solution:
$$2(-2)^2 + 3(5) = 2(4) + 15 = 8 + 15 = 23$$Note: $(-2)^2 = 4$, not $-4$. The exponent applies to the entire base $-2$.
Example 3 — Fraction Expression
Evaluate $\dfrac{n + 3}{2} - n$ when $n = 7$.
Solution:
$$\frac{7 + 3}{2} - 7 = \frac{10}{2} - 7 = 5 - 7 = -2$$Order-of-Operations Reminder (PEMDAS)
- Parentheses
- Exponents
- Multiplication and Division (left to right)
- Addition and Subtraction (left to right)
5.3 Properties of Real Numbers
These properties are laws that hold for all real numbers. Knowing them lets you rearrange and rewrite expressions freely while keeping the value the same.
| Property | Addition Form | Multiplication Form |
|---|---|---|
| Commutative | $a + b = b + a$ | $a \cdot b = b \cdot a$ |
| Associative | $(a+b)+c = a+(b+c)$ | $(ab)c = a(bc)$ |
| Identity | $a + 0 = a$ | $a \cdot 1 = a$ |
| Inverse | $a + (-a) = 0$ | $a \cdot \dfrac{1}{a} = 1$ $(a \neq 0)$ |
| Distributive | $a(b + c) = ab + ac$ | |
| Zero Product | $a \cdot 0 = 0$ | |
The Distributive Property (Expanded)
The distributive property works in both directions and with subtraction too:
$$a(b + c) = ab + ac \qquad a(b - c) = ab - ac$$It also works with more than two terms:
$$a(b + c + d) = ab + ac + ad$$Example 4 — Applying the Distributive Property
Expand $-3(2x - 5)$.
Solution: Multiply $-3$ by each term inside the parentheses.
$$-3(2x - 5) = (-3)(2x) + (-3)(-5) = -6x + 15$$5.4 Like Terms and Combining Them
Definition — Like Terms
Like terms have exactly the same variable part (same variable(s) raised to the same powers). Only the coefficients may differ.
- $3x$ and $-7x$ are like terms (both have variable part $x$).
- $5x^2$ and $2x$ are not like terms (different powers of $x$).
- $4xy$ and $9xy$ are like terms.
- $4xy$ and $4x$ are not like terms ($y$ is missing in the second).
To combine like terms, add or subtract their coefficients and keep the variable part unchanged:
$$5x + 3x = (5+3)x = 8x$$ $$7y^2 - 4y^2 = (7-4)y^2 = 3y^2$$Example 5 — Simplifying by Combining Like Terms
Simplify $4x^2 - 2x + 9 + x^2 + 5x - 3$.
Solution: Group like terms first.
$$= (4x^2 + x^2) + (-2x + 5x) + (9 - 3)$$ $$= 5x^2 + 3x + 6$$Example 6 — Distribute Then Combine
Simplify $3(2x + 1) - 2(x - 4)$.
Solution:
$$= 6x + 3 - 2x + 8 \qquad \text{(distribute each factor)}$$ $$= (6x - 2x) + (3 + 8)$$ $$= 4x + 11$$5.5 Translating Words into Expressions
A major goal of algebra is converting real-world situations into mathematical language. The table below shows common phrases and their algebraic equivalents.
| English Phrase | Operation | Expression (use $n$) |
|---|---|---|
| the sum of $n$ and 8 | addition | $n + 8$ |
| $n$ decreased by 5 | subtraction | $n - 5$ |
| 6 less than $n$ | subtraction | $n - 6$ |
| the product of 4 and $n$ | multiplication | $4n$ |
| $n$ divided by 3 | division | $\dfrac{n}{3}$ |
| twice $n$, increased by 7 | mult. + add. | $2n + 7$ |
| the square of $n$ | exponent | $n^2$ |
| 5 more than three times $n$ | mult. + add. | $3n + 5$ |
Example 7 — Word to Expression
"A rectangle has width $w$ inches. Its length is 4 more than twice its width. Write an expression for the perimeter."
Solution:
- Length: $2w + 4$
- Perimeter of a rectangle: $2(\text{length}) + 2(\text{width})$
5.6 The Negation and Subtraction Connection
Subtracting a quantity is the same as adding its opposite. This is especially important when distributing a negative sign:
$$-(a + b) = -a - b \qquad -(a - b) = -a + b$$Key Skill: Removing Parentheses with a Negative Sign
When a minus sign precedes parentheses, every term inside changes sign:
$$5 - (3x - 2) = 5 - 3x + 2 = 7 - 3x$$Think of it as multiplying by $-1$: $-1(3x - 2) = -3x + 2$.
Practice Problems
Practice — Chapter 5
- Evaluate $5x^2 - 3x + 1$ when $x = -1$.
- Evaluate $\dfrac{2m - n}{3}$ when $m = 4$ and $n = -2$.
- Which property justifies $7 \cdot (x \cdot 3) = 7 \cdot (3 \cdot x)$?
- Use the distributive property to expand $-4(3y - 2)$.
- Simplify: $6a - 3b + 2a + 5b - a$.
- Simplify: $2(x + 4) - 3(2x - 1)$.
- Write an algebraic expression: "7 less than 4 times a number $n$."
- A number $k$ is tripled and then reduced by 9. Write and evaluate the expression when $k = 5$.
Show Answers
- $5(1) - 3(-1) + 1 = 5 + 3 + 1 = 9$
- $\dfrac{2(4)-(-2)}{3} = \dfrac{10}{3} \approx 3.33$
- Commutative Property of Multiplication
- $-12y + 8$
- $(6a+2a-a) + (-3b+5b) = 7a + 2b$
- $2x+8-6x+3 = -4x+11$
- $4n - 7$
- $3k - 9$; when $k=5$: $15-9=6$
Chapter Summary
- A variable represents an unknown; a constant is a fixed value; a coefficient is the number in front of a variable.
- To evaluate an expression, substitute numbers for variables and simplify using PEMDAS.
- The commutative, associative, identity, inverse, and distributive properties allow legal rearrangements of expressions.
- Like terms share the same variable part; combine them by adding or subtracting coefficients.
- To simplify, distribute first, then combine like terms.
- Translate word phrases carefully — watch for "less than" and "more than" which reverse word order.