Math Core

A number like $17$ describes one situation. An expression like $5n + 2$ describes a whole family of situations at once: every value of $n$ gives a different result. Expressions are the language of algebra, so this lesson makes sure you can read them, write them and find their values.

Variables, terms and coefficients

Definition

Variable

A variable is a letter that stands for a number that can change or that you don't know yet. An algebraic expression combines numbers, variables and operations, like $4x - 7$ or $a^2 + 3ab$ . An expression has no equals sign.

A number written right next to a variable means multiplication: $6y$ means $6 \cdot y$ , and $ab$ means $a \cdot b$ . Algebra uses a dot or no symbol at all, because $\times$ is easy to confuse with the letter $x$ .

The parts of an expression that are added or subtracted are its terms. Look at

3m^2 - m + 10.

The terms are $3m^2$ , $-m$ and $10$ . Each term keeps the sign in front of it.
A coefficient is the number multiplying the variable part. The coefficient of $m^2$ is $3$ . The coefficient of $m$ is $-1$ , because $-m$ means $-1 \cdot m$ .
A term with no variable, like $10$ , is a constant term.

Writing expressions from words

Most word phrases turn into symbols in the order you read them. "The product of $8$ and $w$ " is $8w$ . "The sum of $t$ and $5$ , divided by $2$ " is $\dfrac{t + 5}{2}$ .

operation	words that signal it
$+$	sum, plus, more than, increased by, added to
$-$	difference, minus, less than, decreased by, subtracted from
$\cdot$	product, times, twice, triple, of
$\div$	quotient, divided by, per, half of

Two phrases flip the order: "less than" and "subtracted from." "Nine less than $p$ " starts with $p$ and takes $9$ away, so it is $p - 9$ .

Common mistake

Don't write " $3$ less than $x$ " as $3 - x$ . Test it with a number: if $x = 10$ , three less than $10$ is $7$ . That's $10 - 3$ , so the expression is $x - 3$ .

When a phrase says to find a sum or difference first and then do something to it, use parentheses. "Twice the sum of $k$ and $4$ " is $2(k + 4)$ . Without the parentheses, $2k + 4$ means "four more than twice $k$ ," which is different.

Worked example: Translating phrases

Write an expression for each phrase.

Ten more than the quotient of $r$ and $3$ .
Four times the difference of $n$ and $6$ .
Seven subtracted from the square of $y$ .

Solutions.

The quotient of $r$ and $3$ is $\dfrac{r}{3}$ . Ten more than that is $\dfrac{r}{3} + 10$ .
The difference must come first: $4(n - 6)$ .
"Subtracted from" flips the order. Start with $y^2$ and take away $7$ : $y^2 - 7$ .

Evaluating expressions

To evaluate an expression, substitute a value for each variable and simplify with the order of operations. Now that you know integer rules, the values can be negative.

Substitute with parentheses

When you replace a variable with a number, put the number in parentheses. Then follow the order of operations: grouping, exponents, multiplication and division, addition and subtraction.

Parentheses protect the sign. If $x = -5$ , then $x^2 = (-5)^2 = 25$ . But $-x^2$ means "the opposite of $x^2$ ," so $-x^2 = -(-5)^2 = -25$ .

Worked example: Two variables with integers

Evaluate $3a - 2b$ when $a = -4$ and $b = -7$ .

\begin{aligned} 3a - 2b &= 3(-4) - 2(-7) && \text{substitute} \\ &= -12 - (-14) && \text{multiply} \\ &= -12 + 14 && \text{subtracting a negative is adding} \\ &= 2 \end{aligned}

Worked example: Exponents and a fraction bar

Evaluate $\dfrac{x^2 + 1}{x - 3}$ when $x = -2$ .

The fraction bar groups the top and bottom, so simplify each one first.

Top: $(-2)^2 + 1 = 4 + 1 = 5$ .
Bottom: $-2 - 3 = -5$ .

So the value is $\dfrac{5}{-5} = -1$ .

Expressions that model situations

An expression can describe a real rule. A bike rental shop charges a $6 helmet fee plus $4 per hour. For $h$ hours, the hourly part costs $4h$ dollars, so the total is $4h + 6$ dollars. For a $3$ -hour ride, substitute $h = 3$ : $4(3) + 6 = 18$ , so the ride costs $18.

Tip

When you build an expression from a situation, try it with a small number you can check in your head. If $h = 1$ , the rental should cost $4 plus $6, which is $10, and $4(1) + 6 = 10$ .

Practice

Practice 1

How many terms does the expression $4a^2 + 9 - 3a + b$ have?

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

Practice 2

What is the constant term of $3m - 8 + m^2$ ?

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

Practice 3

Which expression means "twelve less than twice a number $k$ "?

Practice 4

Write an expression for "half of the difference of $p$ and $10$ ."

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

Practice 5

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

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

Practice 6

Evaluate $-x^2 + 3x$ when $x = -4$ .

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

Practice 7

Evaluate $\dfrac{3m - n}{m + n}$ when $m = 4$ and $n = -2$ .

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

Practice 8

A phone plan costs $25 per month plus $0.10 for each text message. Write an expression for the monthly cost with $t$ texts, then find the cost in dollars for a month with $150$ texts.

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