Use the Language of Algebra

Use Variables and Algebraic Symbols

Suppose this year Greg is $20$ years old and Alex is $23$. Alex is always $3$ 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 is always $3$ more. In the language of algebra, we say that Greg’s age and Alex’s age are variables and the $3$ is a constant. The ages change (“vary”) but the $3$ years between them always stays the same (“constant”).

In algebra, we use letters of the alphabet to represent variables. If we call Greg’s age $g$, then Alex’s age is $g + 3$.

The letters most commonly used for variables are $x$, $y$, $a$, $b$, and $c$. To write algebraically, we need operation symbols as well as numbers and variables.

When translating between symbols and English, pay attention to the words “of” and “and”: the difference of $9$ and $2$ means $9 - 2$; the product of $4$ and $8$ means $4 \cdot 8$. In algebra, the cross symbol $\times$ is not used for multiplication, because it can be confused with the variable $x$ — does $3xy$ mean $3 \times y$ (“three times $y$”) or $3 \cdot x \cdot y$ (“three times $x$ times $y$”)? Using $\cdot$ or parentheses keeps it clear.

When two quantities have the same value, we connect them with an equal sign: $a = b$ is read “$a$ is equal to $b$.”

On the number line, numbers get larger from left to right, which explains the symbols $<$ and $>$: $a < b$ ("$a$ is less than $b$") means $a$ is to the left of $b$ on the number line, and $a > b$ ("$a$ is greater than $b$") means $a$ is to the right of $b$. The statement $a < b$ is equivalent to $b > a$ — for example, $7 < 11$ is equivalent to $11 > 7$.

Example. Translate from algebra into English: (a) $17 \le 26$ (b) $8 \neq 17 - 3$ (c) $12 > 27 \div 3$ (d) $y + 7 < 19$.

(a) $17$ is less than or equal to $26$. (b) $8$ is not equal to $17$ minus $3$. (c) $12$ is greater than $27$ divided by $3$. (d) $y$ plus $7$ is less than $19$.

Grouping symbols — parentheses $(\ )$, brackets $[\ ]$, and braces $\{\ \}$ — work like punctuation marks in English: they show which expressions belong together. For example, $8(14 - 8)$, $21 - 3[2 + 4(9 - 8)]$, and $24 \div \{13 - 2[1(6-5) + 4]\}$ all use grouping symbols; we’ll simplify expressions like these later in this section.

A phrase expresses a single thought that is incomplete by itself, but a sentence makes a complete statement with a subject and a verb. In algebra, an expression is like a phrase, and an equation is like a sentence.

For example, $3 + 5$ ("$3$ plus $5$," the sum of three and five) is an expression, while $3 + 5 = 8$ (“the sum of three and five is equal to eight”) is an equation — the equal sign gives it a verb, making it a complete sentence.

Example. Determine if each is an expression or an equation: (a) $2(x + 3) = 10$ (b) $4(y - 1) + 1$ (c) $x \div 25$ (d) $y + 8 = 40$.

(a) is an equation — two expressions are connected with an equal sign. (b) is an expression — no equal sign. (c) is an expression — no equal sign. (d) is an equation — two expressions are connected with an equal sign.

Suppose we need to multiply $2$ by itself nine times. Writing $2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2$ is tedious, so instead we use exponents: $2 \cdot 2 \cdot 2$ is written $2^3$, and $2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2$ is written $2^9$. In an expression like $2^3$, the $2$ is called the base and the $3$ is the exponent — it tells us how many times to multiply the base by itself. We read $2^3$ as “two to the third power” or “two cubed.”

While we read $a^n$ as “$a$ to the $n^{\text{th}}$ power,” we usually read $a^2$ as “$a$ squared” and $a^3$ as “$a$ cubed” — you’ll see later why these two have special names.

Example. Simplify $3^4$. Expand: $3 \cdot 3 \cdot 3 \cdot 3$. Multiply left to right: $9 \cdot 3 \cdot 3 = 27 \cdot 3 = 81$.

Simplify Expressions Using the Order of Operations

To simplify an expression means to do all the operations in it. For example, to simplify $4 \cdot 2 + 1$, we multiply first to get $8$, then add $1$ to get $9$. A good habit is to work down the page, writing each step below the previous one, and never use an equal sign when simplifying an expression — that helps avoid confusing expressions with equations.

Consider $4 + 3 \cdot 7$. Some students say $49$ (add first, then multiply: $7 \cdot 7 = 49$); others say $25$ (multiply first, then add: $4 + 21 = 25$). The same expression should always give the same result, so mathematicians established the Order of Operations:

A common mnemonic takes the first letter of each step: “Please Excuse My Dear Aunt Sally” (Parentheses, Exponents, Multiplication/Division, Addition/Subtraction). “My Dear” reminds us that multiplication and division have equal priority — we don’t always multiply before dividing or always divide before multiplying, we work left to right. Likewise “Aunt Sally” reminds us that addition and subtraction have equal priority and are also done left to right.

Example. Simplify (a) $4 + 3 \cdot 7$ (b) $(4+3) \cdot 7$.

(a) No parentheses, no exponents, but there is multiplication: multiply first to get $4 + 21$, then add to get $25$.

(b) There are parentheses: simplify inside them first to get $(7) \cdot 7$, then multiply to get $49$. Notice grouping symbols change the result.

Example. Simplify $18 \div 6 + 4(5-2)$.

Parentheses first: $18 \div 6 + 4(3)$. No exponents. Multiplication and division, left to right — divide first since it comes first: $3 + 4(3)$, then multiply: $3 + 12$. Finally add: $15$.

Example. Simplify $5 + 2^3 + 3[6 - 3(4-2)]$.

When there are multiple grouping symbols, simplify the innermost first and work outward. Focus on the parentheses inside the brackets: $4 - 2 = 2$, giving $5 + 2^3 + 3[6 - 3(2)]$. Continue inside the brackets: multiply $3(2) = 6$, giving $5 + 2^3 + 3[6-6]$, then subtract to get $5 + 2^3 + 3[0]$ — the brackets need no further simplification. Now the exponent: $2^3 = 8$, giving $5 + 8 + 3[0]$. Multiply: $5 + 8 + 0$. Add: $13 + 0 = 13$.

Evaluate an Expression

To evaluate an expression means to find its value when the variable is replaced by a given number. Substitute the number for the variable, then follow the order of operations to simplify.

Example. Evaluate $7x - 4$ when (a) $x = 5$ and (b) $x = 1$.

(a) $7(5) - 4 = 35 - 4 = 31$. (b) $7(1) - 4 = 7 - 4 = 3$.

Example. Evaluate for $x = 4$: (a) $x^2$ (b) $3^x$.

(a) Replace $x$ with $4$: $4^2 = 4 \cdot 4 = 16$.

(b) Replace $x$ with $4$: $3^4 = 3 \cdot 3 \cdot 3 \cdot 3 = 81$.

Example. Evaluate $2x^2 + 3x + 8$ when $x = 4$.

Substitute $x = 4$: $2(4)^2 + 3(4) + 8$. Follow the order of operations — exponent first: $2(16) + 3(4) + 8$. Multiply: $32 + 12 + 8$. Add: $52$.

Identify and Combine Like Terms

Algebraic expressions are made up of terms. A term is a constant, or the product of a constant and one or more variables — examples include $7$, $y$, $5x^2$, $9a$, and $b^5$. The constant that multiplies the variable is called the coefficient. Think of the coefficient as the number in front of the variable: the coefficient of $3x$ is $3$. When we write $x$ alone, the coefficient is $1$, since $x = 1 \cdot x$.

Example. Identify the coefficient of each: (a) $14y$ (b) $15x^2$ (c) $a$.

(a) The coefficient of $14y$ is $14$. (b) The coefficient of $15x^2$ is $15$. (c) The coefficient of $a$ is $1$, since $a = 1 \cdot a$.

Some terms share common traits. Consider the six terms $5x$, $7$, $n^2$, $4$, $3x$, $9n^2$: $7$ and $4$ are both constants; $5x$ and $3x$ are both terms with $x$; $n^2$ and $9n^2$ are both terms with $n^2$. When two terms are constants, or have the same variable raised to the same power, we say they are like terms.

Example. Identify the like terms: $y^3, 7x^2, 14, 23, 4y^3, 9x, 5x^2$.

$y^3$ and $4y^3$ are like terms — same variable and exponent. $7x^2$ and $5x^2$ are like terms — same variable and exponent. $14$ and $23$ are like terms — both constants. There is no other term like $9x$.

Adding or subtracting terms forms an expression. In $2x^2 + 3x + 8$, the three terms are $2x^2$, $3x$, and $8$.

Example. Identify the terms in each expression: (a) $9x^2 + 7x + 12$ (b) $8x + 3y$.

(a) The terms are $9x^2$, $7x$, and $12$. (b) The terms are $8x$ and $3y$.

If there are like terms in an expression, we can simplify by combining them. What would $4x + 7x + x$ simplify to? Thinking of it as $4$ of something, plus $7$ more, plus $1$ more, gives $12$ of that thing — add the coefficients and keep the same variable, regardless of what $x$ represents: $4x + 7x + x = 12x$.

Example. Simplify $2x^2 + 3x + 7 + x^2 + 4x + 5$.

Identify the like terms — $2x^2$ and $x^2$; $3x$ and $4x$; $7$ and $5$. Rearrange so like terms are together: $2x^2 + x^2 + 3x + 4x + 7 + 5$. Combine: $3x^2 + 7x + 12$.

Translate an English Phrase to an Algebraic Expression

Now we’ll reverse the earlier process and translate English phrases into algebraic expressions.

Each phrase involving two numbers tells us to operate on them, and the words “of” and “and” mark where the numbers are.

Example. Translate each phrase into an algebraic expression: (a) the difference of $17x$ and $5$ (b) the quotient of $10x^2$ and $7$.

(a) Difference means subtraction; look for “of” and “and” to find the numbers to subtract: $17x - 5$.

(b) Quotient means division: $10x^2 \div 7$, also written $10x^2/7$ or $\tfrac{10x^2}{7}$.

How old will you be in eight years? What age is $8$ more than your age now? “Eight more than” means $8$ added to your present age. How old were you seven years ago? “Seven less than” means $7$ subtracted from your present age.

Example. Translate into an algebraic expression: (a) seventeen more than $y$ (b) nine less than $9x^2$.

(a) More than means addition, and “more than” means “added to”: seventeen more than $y$ is seventeen added to $y$, so $y + 17$.

(b) Less than means subtraction, and “less than” means “subtracted from”: nine less than $9x^2$ is nine subtracted from $9x^2$, so $9x^2 - 9$.

Example. Translate into an algebraic expression: (a) five times the sum of $m$ and $n$ (b) the sum of five times $m$ and $n$.

(a) Times means multiply, sum means add. Because we’re multiplying $5$ by the whole sum, we need parentheses around $m + n$ to force the sum first (order of operations): $5(m+n)$.

(b) Here we’re taking the sum of “five times $m$” and $n$ — look for “of” and “and” to see what’s being added: $5m + n$. Notice how the parentheses in (a) change the meaning.

Later in this course, we’ll apply algebra to solving applications, and the first step is translating an English phrase into an algebraic expression.

Example. The length of a rectangle is $6$ less than the width. Let $w$ represent the width. Write an expression for the length.

A phrase about the length: “$6$ less than the width.” Substitute $w$ for “the width”: “$6$ less than $w$.” Rewrite “less than” as “subtracted from”: “$6$ subtracted from $w$.” Translate: $w - 6$.

Example. June has dimes and quarters in her purse. The number of dimes is three less than four times the number of quarters. Let $q$ represent the number of quarters. Write an expression for the number of dimes.

A phrase about the number of dimes: “three less than four times the number of quarters.” Substitute $q$ for “the number of quarters”: “three less than $4$ times $q$.” Translate “$4$ times $q$” as $4q$: “three less than $4q$.” Translate the phrase into algebra: $4q - 3$.

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 numbers and variables using operation symbols. equation — two expressions connected by an equal sign. base and exponent — in $a^n$, the base $a$ is the number being repeatedly multiplied, and the exponent $n$ tells how many times. term — a constant, or the product of a constant and one or more variables. coefficient — the constant that multiplies the variable in a term. like terms — terms that are either constants or have the same variables raised to the same powers.

This section is adapted from Elementary Algebra 2e, Section 1.2: 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 the narrative examples and tables of notation; omitted the Be Prepared quiz, Manipulative Mathematics callout, Self Check checklist, and end-of-section exercises; and converted the practice problems (“Try Its”) into interactive exercises with instant feedback.