Lesson 1: Use the Language of Algebra

New Concept

Algebra is a language for describing relationships. This lesson introduces its basic vocabulary: variables (letters for changing values) and constants (fixed numbers). You will learn to form expressions and equations using symbols, all governed by the order of operations.

What’s next

Now that you have the basics, get ready for interactive examples and practice cards to master simplifying expressions using the order of operations.

Property

A variable is a letter that represents a number or quantity whose value may change.
A constant is a number whose value always stays the same.
In algebra, letters of the alphabet are used to represent variables. Letters often used for variables are $x$, $y$, $a$, $b$, and $c$.

Examples

• In the expression $c+5$, $c$ is the variable because its value can change, and $5$ is the constant because its value is fixed.
• If a movie ticket costs 8 dollars, the number of tickets you buy is a variable ($t$), but the price of each ticket is a constant (8 dollars).
• Sarah is 5 years younger than her brother, Mark. If Mark's age is $m$, Sarah's age is $m-5$. Here, $m$ is a variable and $5$ is a constant.

Explanation

In algebra, we use letters (variables) for numbers that can change, like your height. Numbers that always stay the same, like the number of days in a week, are constants. This helps us write rules for changing situations.

Property

An inequality is used in algebra to compare two quantities that may have different values. We use the symbols '$<$' and '$>$' for inequalities.
$a = b$ is read $a$ is equal to $b$
$a \neq b$ is read $a$ is not equal to $b$
$a < b$ is read $a$ is less than $b$
$a > b$ is read $a$ is greater than $b$
$a \leq b$ is read $a$ is less than or equal to $b$
$a \geq b$ is read $a$ is greater than or equal to $b$

Examples

• The statement '15 is greater than 9' is written in algebra as $15 > 9$.
• To show that your age, $a$, must be at least 18 to vote, you would write $a \geq 18$.
• The expression $7 \neq 4+4$ means 'seven is not equal to the sum of four and four'.

Explanation

These symbols are the verbs of math sentences. They tell us the relationship between two values: whether they're equal, not equal, or if one is bigger or smaller than the other. The small point of $<$ or $>$ always faces the smaller number.

Property

An expression is a number, a variable, or a combination of numbers and variables and operation symbols.
An equation is made up of two expressions connected by an equal sign.

Examples

• $7x - 2$ is an expression because it does not have an equal sign and does not state a complete relationship.
• $4 + 9 = 13$ is an equation because it links two expressions ($4+9$ and $13$) with an equal sign.
• The phrase 'the quotient of $y$ and 3' is an expression written as $\frac{y}{3}$, while the sentence 'the quotient of $y$ and 3 is 6' is an equation written as $\frac{y}{3} = 6$.

Explanation

Think of it like language. An expression is a math phrase, like 'x plus three', which is an incomplete thought. An equation is a full sentence, like 'x plus three equals five', because the equals sign connects two expressions to make a complete statement.

Property

For any expression $a^n$, $a$ is a factor multiplied by itself $n$ times if $n$ is a positive integer.
The expression $a^n$ is read $a$ to the $n^{th}$ power. In $a^n$, the $a$ is called the base and the $n$ is called the exponent.
$a^n = a \cdot a \cdot a \cdot ... \cdot a$ ($n$ factors)
Special names are used for powers of 2 and 3:
$a^2$ is read as '$a$ squared'
$a^3$ is read as '$a$ cubed'

Examples

• The expression $5 \cdot 5 \cdot 5 \cdot 5$ can be written in exponential notation as $5^4$, where 5 is the base and 4 is the exponent.
• To write $y^3$ in expanded form, you write out the base $y$ multiplied by itself 3 times: $y \cdot y \cdot y$.
• To simplify $2^5$, you calculate $2 \cdot 2 \cdot 2 \cdot 2 \cdot 2$, which equals $32$.

Explanation

Exponents are a shortcut for writing repeated multiplication. The small exponent number tells you how many times to multiply the larger base number by itself. This makes it much faster to write out long calculations involving the same factor.

Property

When simplifying mathematical expressions perform the operations in the following order:

1. Parentheses and other Grouping Symbols: Simplify all expressions inside parentheses or other grouping symbols, working on the innermost parentheses first.
2. Exponents: Simplify all expressions with exponents.
3. Multiplication and Division: Perform all multiplication and division in order from left to right. These operations have equal priority.
4. Addition and Subtraction: Perform all addition and subtraction in order from left to right. These operations have equal priority.

A common way to remember this is the phrase 'Please Excuse My Dear Aunt Sally'.

Examples

• To simplify $10 - 2 \cdot 3$, you perform multiplication before subtraction: $10 - 6 = 4$.
• To simplify $(10 - 2) \cdot 3$, you perform the operation in parentheses first: $8 \cdot 3 = 24$.
• To simplify $5 + (4-1)^2 \div 3$, you start with parentheses ($5+3^2 \div 3$), then exponents ($5+9 \div 3$), then division ($5+3$), and finally addition, which gives $8$.

Explanation

This is the official rulebook for solving math problems. Following this order (PEMDAS) ensures everyone gets the same correct answer. Always handle groups first, then powers, then multiplication/division, and finally addition/subtraction from left to right.