Use the Language of Algebra

Use variables and algebraic symbols

Greg and Alex have the same birthday, but they were born in different years. This year Greg is $20$ years old and Alex is $23$, so Alex is $3$ years older than Greg. When Greg was $12$, Alex was $15$. When Greg is $35$, Alex will be $38$. No matter what Greg’s age is, Alex’s age will always be $3$ years more.

In the language of algebra, we say that Greg’s age and Alex’s age are variable and the three is a constant. The ages change, or vary, so age is a variable. The $3$ years between them always stays the same, so the age difference is the constant. In algebra, letters of the alphabet are used to represent variables. If we call Greg’s age $g$, then we can use $g + 3$ to represent Alex’s age.

Letters often used for variables are $x, y, a, b,$ and $c$.

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.

To write algebraically, we need symbols as well as numbers and variables. The four basic operations are written like this:

In algebra, the cross symbol $\times$ is not used to show multiplication because it can be confused with the variable $x$. To make multiplication clear, use a center dot $\cdot$ or parentheses.

When translating between symbols and words, pay attention to the words of and and to find the numbers. The sum of $5$ and $3$ means add $5$ plus $3$, which we write as $5 + 3$. The product of $4$ and $8$ means multiply, which we write as $4 \cdot 8$.

Example. Translate from algebra to words: (a) $12 + 14$, (b) $(30)(5)$, (c) $64 \div 8$, (d) $x - y$.

(a) the sum of twelve and fourteen; (b) the product of thirty and five; (c) the quotient of sixty-four and eight; (d) the difference of $x$ and $y$.

When two quantities have the same value, we say they are equal and connect them with an equal sign, $=$. An inequality compares two quantities that may have different values. On the number line, numbers get larger from left to right, so if $b$ is greater than $a$, then $b$ is to the right of $a$. We write $a < b$ ("$a$ is less than $b$") and $a > b$ ("$a$ is greater than $b$").

The expression $a < b$ can be read left-to-right or right-to-left, so $a < b$ is equivalent to $b > a$. For example, $7 < 11$ is equivalent to $11 > 7$. A line under the symbol, as in $a \le b$, means “$a$ is less than or equal to $b$”, and a slash through the equal sign, $\ne$, means “not equal.”

The symbols $<$ and $>$ each have a smaller side and a larger side. The smaller side faces the smaller number, and the larger side faces the larger number.

Example. Compare the fuel economy, in miles per gallon, of five cars:

Translate each comparison into algebraic notation: the Prius’s MPG is greater than the Mini Cooper’s, $48 > 27$; the Corolla’s MPG is greater than the Versa’s, $28 > 26$; the Fit’s MPG is less than the Prius’s, $33 < 48$; the Mini Cooper’s MPG is not equal to the Fit’s, $27 \ne 33$.

Grouping symbols — parentheses $(\ )$, brackets $[\ ]$, and braces $\{\ \}$ — work like punctuation. They show which parts of an expression are kept together and simplified as a unit.

Identify expressions and equations

In English, a phrase expresses a single incomplete thought, while a sentence makes a complete statement with a subject and a verb. Algebra has a parallel distinction.

An expression is like a phrase: a number, a variable, or a combination of numbers, variables, and operation symbols, such as $3 + 5$, $y - 1$, or $6 \cdot 7$. An equation is two expressions connected by an equal sign — the equal sign acts as the verb, making a complete sentence. For example, $3 + 5 = 8$ reads “the sum of three and five is equal to eight.”

Example. Determine if each is an expression or an equation: $16 - 6 = 10$ is an equation (two expressions joined by an equal sign); $4 \cdot 2 + 1$ is an expression (no equal sign); $x \div 25$ is an expression; $y + 8 = 40$ is an equation.

Simplify expressions with exponents

To simplify a numerical expression means to do all the math possible. For example, to simplify $4 \cdot 2 + 1$ we first multiply $4 \cdot 2$ to get $8$, then add $1$ to get $9$.

Suppose we have $2 \cdot 2 \cdot 2$. We can write this more compactly using exponential notation as $2^3$. Here the $2$ is the base and the $3$ is the exponent; the exponent tells how many factors of the base to multiply. We say $2^3$ is in exponential notation and $2 \cdot 2 \cdot 2$ is in expanded notation.

For any positive integer $n$, the notation $a^n$ means $a$ is a factor multiplied by itself $n$ times:

The expression $a^n$ is read “$a$ to the $n$th power.” Two powers have special names: $a^2$ is read “$a$ squared” and $a^3$ is read “$a$ cubed.”

Example. Write $16 \cdot 16 \cdot 16 \cdot 16 \cdot 16 \cdot 16 \cdot 16$ in exponential form. The base $16$ is a factor $7$ times, so this is $16^7$.

Example. Write $8^6$ in expanded form. The base is $8$ and the exponent is $6$, so $8^6$ means $8 \cdot 8 \cdot 8 \cdot 8 \cdot 8 \cdot 8$.

To simplify an exponential expression without a calculator, write it in expanded form and multiply the factors. For example, $3^4 = 3 \cdot 3 \cdot 3 \cdot 3 = 81$.

Simplify expressions using the order of operations

Consider the expression $4 + 3 \cdot 7$. If we add first we get $49$; if we multiply first we get $25$. The same expression should give only one result, so mathematicians agreed on the order of operations.

A common way to remember the order is the phrase Please Excuse My Dear Aunt Sally (Parentheses, Exponents, Multiplication/Division, Addition/Subtraction). Because “My Dear” and “Aunt Sally” each pair up, multiplication and division have equal priority and are done left to right — and likewise for addition and subtraction.

Example. Simplify $4 + 3 \cdot 7$. There are no parentheses or exponents, so multiply first: $4 + 21$, then add: $25$.

Example. Simplify $(4 + 3) \cdot 7$. The parentheses come first: $(7) \cdot 7$, then multiply: $49$. The grouping symbols change the result.

Multiplication and division are done in order from left to right — neither one always comes first.

Example. Simplify (a) $18 \div 9 \cdot 2$ and (b) $18 \cdot 9 \div 2$. Both have only multiplication and division, so work strictly left to right in each: (a) divide first, $2 \cdot 2 = 4$; (b) multiply first, $162 \div 2 = 81$. Even though the same numbers and operations appear in a different order, left-to-right order gives two different — and correct — results.

Example. Simplify $42 \div 7 \cdot 3$. Working left to right, divide first: $6 \cdot 3$, then multiply: $18$.

Example. Simplify $18 \div 6 + 4(5 - 2)$. Parentheses first: $18 \div 6 + 4(3)$. Then multiplication and division left to right: $3 + 4(3) = 3 + 12$. Finally add: $15$.

When an expression has more than one type of grouping symbol — parentheses inside brackets, for instance — work the innermost grouping symbol first, then move outward.

Example. Simplify $5 + 2^3 + 3[6 - 3(4 - 2)]$. Work the innermost parentheses first: $4 - 2 = 2$, giving $5 + 2^3 + 3[6 - 3 \cdot 2]$. Inside the brackets, multiply before subtracting: $3 \cdot 2 = 6$, so $6 - 6 = 0$, leaving $5 + 2^3 + 3[0]$. Now the brackets are gone: simplify the exponent, $2^3 = 8$, giving $5 + 8 + 3 \cdot 0$. Multiply: $3 \cdot 0 = 0$. Finally add left to right: $5 + 8 + 0 = 13$.

When several exponents appear, they may be simplified in the same step.

Example. Simplify $2^3 + 3^4 \div 3 - 5^2$. Exponents first: $8 + 81 \div 3 - 25$. Then divide: $8 + 27 - 25$. Then add and subtract left to right: $35 - 25 = 10$.

Key terms

variable — a letter that represents a number whose value may change. constant — a number whose value always stays the same. expression — a number, a variable, or a combination of these with operation symbols. equation — two expressions connected by an equal sign. base — in $a^n$, the factor $a$ being repeated. exponent — in $a^n$, the number $n$ of factors of the base. order of operations — the agreed order for simplifying: parentheses, exponents, multiplication and division (left to right), then addition and subtraction (left to right).

This section is adapted from Prealgebra 2e, Section 2.1: Use the Language of Algebra by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: condensed prose, recreated tables in accessible Markdown, and converted practice problems (“Try Its”) into interactive exercises with instant feedback.