Math Core

Arithmetic works with specific numbers. Algebra lets you write one rule that works for every number at once, like "the cost is $3 per ticket plus a $2 fee" for any number of tickets. The tool that makes this possible is the variable.

Variables and constants

Definition

Variable

A variable is a letter or symbol that stands for a number that can change or that you don't know yet. A constant is a number whose value never changes, like $7$ or $-2.5$ .

Any letter can be a variable. Choosing a letter that reminds you of the quantity helps: $t$ for time, $c$ for cost, $n$ for "number of."

When a number and a variable are written side by side, they are multiplied. So $4n$ means $4 \times n$ , and $xy$ means $x \times y$ . You will also see a raised dot, $4 \cdot n$ , or parentheses, $4(n)$ . Algebra avoids the $\times$ symbol because it looks too much like the letter $x$ .

Parts of an expression

An algebraic expression is a combination of numbers, variables and operations, such as

5x^2 - 3y + 8.

The pieces of an expression that are added or subtracted are called terms. This expression has three terms: $5x^2$ , $-3y$ and $8$ .

word	meaning	in $5x^2 - 3y + 8$
term	a piece separated by $+$ or $-$	$5x^2$ , $-3y$ , $8$
coefficient	the number multiplying a variable	$5$ and $-3$
constant term	a term with no variable	$8$

Notice that the minus sign belongs to the term after it. The coefficient of $y$ is $-3$ , not $3$ . And when no number is written in front of a variable, the coefficient is $1$ : in $x + 4$ , the coefficient of $x$ is $1$ . In $-x$ , it is $-1$ .

Common mistake

An expression is not an equation. An expression like $2x + 1$ has no equals sign; you can simplify it or evaluate it, but you can't "solve" it. An equation like $2x + 1 = 9$ states that two expressions are equal, and that is what you solve (in the next unit).

Translating words into expressions

Word problems use many phrases for the four operations. Here are the most common.

operation	phrases
addition	sum, plus, more than, increased by, total
subtraction	difference, minus, less than, decreased by, fewer than
multiplication	product, times, of, twice (times 2), triple (times 3)
division	quotient, divided by, per, ratio, half of (divided by 2)

Most phrases translate in the order you read them: "the product of $6$ and $n$ " is $6n$ . But "less than" and "subtracted from" reverse the order. "Four less than $n$ " means start with $n$ and take away $4$ , so it is $n - 4$ , not $4 - n$ .

Tip

Test a translation with a real number. If $n = 10$ , "four less than $n$ " should be $6$ . Since $10 - 4 = 6$ and $4 - 10 = -6$ , the expression $n - 4$ is the right one.

Worked example: Translating phrases

Write an expression for each phrase.

Seven more than twice a number $x$ .
The quotient of a number $m$ and $5$ , decreased by $3$ .
Three times the sum of $y$ and $2$ .

Solutions.

"Twice a number" is $2x$ , and "seven more than" adds $7$ : $2x + 7$ .
"The quotient of $m$ and $5$ " is $\dfrac{m}{5}$ . Decreasing by $3$ gives $\dfrac{m}{5} - 3$ .
"The sum of $y$ and $2$ " must be computed first, so it needs parentheses: $3(y + 2)$ . Without them, $3y + 2$ would mean "two more than three times $y$ ," a different quantity.

Evaluating expressions

To evaluate an expression, replace each variable with its given value and then simplify using the order of operations. This is called substitution.

How to evaluate an expression

Rewrite the expression, putting each value in parentheses where its variable was.
Simplify using the order of operations.

The parentheses matter most when a value is negative. If $x = -3$ , then $x^2$ becomes $(-3)^2 = 9$ . Writing $-3^2$ instead would give $-9$ , which is wrong.

Worked example: Evaluating with two variables

Evaluate $4a - 2b$ when $a = 6$ and $b = 5$ .

\begin{aligned} 4a - 2b &= 4(6) - 2(5) && \text{substitute} \\ &= 24 - 10 && \text{multiply} \\ &= 14 && \text{subtract} \end{aligned}

Worked example: Evaluating with a negative value

Evaluate $x^2 - 5x + 1$ when $x = -2$ .

\begin{aligned} x^2 - 5x + 1 &= (-2)^2 - 5(-2) + 1 && \text{substitute} \\ &= 4 - 5(-2) + 1 && \text{exponent} \\ &= 4 + 10 + 1 && \text{multiply: } -5 \cdot (-2) = 10 \\ &= 15 && \text{add} \end{aligned}

Worked example: An expression that models a situation

A streaming service charges a $4 sign-up fee plus $9 per month. Write an expression for the total cost of $m$ months, then find the cost of one year.

Each month adds $9, so $m$ months cost $9m$ dollars. Adding the one-time fee gives $9m + 4$ .

One year is $12$ months: $9(12) + 4 = 108 + 4 = 112$ . A year costs $112.

Practice

Practice 1

How many terms does the expression $6x^2 - x + 11$ have?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 2

What is the coefficient of $x$ in $6x^2 - x + 11$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 3

Which expression means "five less than a number $n$ "?

Practice 4

Write an expression for "three times the sum of a number $n$ and $8$ ."

Enter an expression, e.g. 3x^2 - 2x + 1

Practice 5

Evaluate $5p + 2q$ when $p = 3$ and $q = 8$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 6

Evaluate $2x^2 - 3x + 1$ when $x = -3$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 7

Evaluate $\dfrac{a + b}{a - b}$ when $a = 10$ and $b = 6$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 8

A gym charges a $15 joining fee plus $8 per visit. Write an expression for the cost of $v$ visits, then find the total cost of $4$ visits in dollars.

Enter a number. Fractions like 3/4 and sqrt(2) are OK.