Variables and Expressions

Learning goals

Use a letter for a number that is unknown or free to change
Tell an expression from an equation, and say what you do to each
Translate a phrase into algebra, minding the reversal in less than
Name the terms, coefficient and constant of an expression
Evaluate by substituting each value in parentheses

What a variable is

A variable is a letter that stands for a number. The letter does not have a fixed value of its own; it is a placeholder, exactly the way a blank box would be. Writing

3 \times n

says “three times whatever number $n$ is.” If $n$ is $4$ , the phrase is worth $3 \times 4 = 12$ ; if $n$ is $10$ , it is worth $3 \times 10 = 30$ . The same letter can take different values on different occasions, which is precisely why it is called variable: its value can vary.

There are two everyday reasons to reach for a letter instead of a number.

The first is that the number is unknown. You may be trying to find it, the way you will in the next few lessons when you solve for $x$ . Until you pin it down, a letter holds its place so you can still write down everything you do know about it.

The second is that the number is changing, or you want a single statement to cover many cases at once. The cost of $n$ notebooks is $3 \times n$ for every possible $n$ ; you do not want to rewrite the rule for each separate purchase. One expression with a variable replaces an endless list of arithmetic examples.

Any letter will do, and the choice is up to you. People lean on $x$ and $y$ out of long habit. Even so, a letter that hints at its meaning often reads better: $t$ for time, $d$ for distance, $n$ for a count of things. The letter is just a name; what matters is that you state clearly what it stands for.

Expressions versus equations

These two words name different things, and keeping them apart will save you constant confusion later.

An algebraic expression is a combination of numbers, variables, and operations ( $+$ , $-$ , $\times$ , $\div$ , exponents). Such a combination stands for a single value once you know the variables. It is a phrase, the algebraic version of a number. Each of these is an expression:

3n, \qquad x + 5, \qquad 2a - 7, \qquad \frac{y}{4}, \qquad 5x^2.

An expression has no equals sign. On its own it makes no claim; it simply names an amount, the way $3 \times 4$ names an amount before you work it out.

An equation is two expressions joined by an equals sign, stating that they have the same value. It is a complete sentence, and like any sentence it can be true or false. Each of these is an equation:

3n = 12, \qquad x + 5 = 9, \qquad 2a - 7 = a.

The difference is exactly the difference between a phrase and a sentence. “Three more than $x$ ” is a phrase; it points at a quantity but asserts nothing. “Three more than $x$ equals nine” is a sentence; it makes a claim you could check. You evaluate an expression (work out its value), but you solve an equation (find the variable value that makes the sentence true). Solving is the work of the coming lessons; this lesson is about reading and evaluating expressions.

An expression is a phrase that names a value; adding an equals sign and a second expression makes an equation, a full sentence that can be true or false.

Check your understanding

Which of these is an equation (not just an expression)?

7x - 2

4a + 3 = 11

\frac{m}{6}

x^2 + x

Translating words into expressions

Most of the time a problem arrives in words, and the first real skill of algebra is turning those words into an expression. The reliable method is to name the unknown with a letter, then replace each piece of the sentence with its arithmetic operation. A small vocabulary covers almost everything:

Words	Operation	Example phrase	Expression
sum, more than, increased by, total	$+$	seven more than $x$	$x + 7$
difference, less than, decreased by, fewer	$-$	five less than $x$	$x - 5$
product, times, of, twice, double	$\times$	three times $x$	$3x$
quotient, divided by, per, split	$\div$	$x$ divided by $4$	$\dfrac{x}{4}$

Two of these phrases hide a trap worth pausing on. “Five less than $x$ ” is $x - 5$ , not $5 - x$ . The phrase tells you to start with $x$ and take five away, so the $x$ comes first even though $5$ is spoken first. The same flip happens with “fewer than” and “subtracted from.” Read these by their meaning, not by the left-to-right order of the words.

When several operations combine, build the expression one phrase at a time, and use parentheses whenever a whole group must be operated on together.

Worked example 1 Translate three phrases into expressions

Name the unknown number $x$ in each, then convert phrase by phrase.

“Eight more than twice a number.” Twice the number is $2x$ , and eight more than that adds $8$ :

2x + 8.

“The product of $5$ and a number, decreased by $3$ .” The product of $5$ and the number is $5x$ , and decreased by $3$ subtracts $3$ :

5x - 3.

“A number divided by $2$ , then increased by $1$ .” Dividing by $2$ gives $\dfrac{x}{2}$ , and increasing by $1$ adds $1$ :

\frac{x}{2} + 1.

In each case the unknown is held by the letter $x$ , and every word maps to one operation.

The trickiest translations are the ones where a sum or difference must be operated on as a whole. Here parentheses do the grouping, exactly as they did in the order-of-operations chapter.

Worked example 2 Translate 'twice the sum of a number and four'

The phrase has two layers. The inner phrase, “the sum of a number and four,” is $x + 4$ . The outer word “twice” then doubles that entire sum, not just the $x$ .

To double the whole sum, wrap it in parentheses before multiplying:

2(x + 4).

Without the parentheses, $2x + 4$ would double only the $x$ and leave the $4$ alone, a different expression. Compare the two by trying $x = 3$ : the correct $2(3 + 4) = 14$ , but $2x + 4$ would give $2 \cdot 3 + 4 = 10$ . The parentheses are what keep “twice the sum” honest.

Check your understanding

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

4 - n

n - 4

4n

n + 4

The parts of an expression

To work with an expression you need names for its pieces. Consider

5x + 3y - 7.

The expression is built from parts that are added or subtracted together; each such part is a term. This expression has three terms: $5x$ , $3y$ , and $-7$ . Terms are the chunks separated by the $+$ and $-$ signs. By convention a term carries the sign written in front of it, so the subtracted piece is the term $-7$ , not a bare $7$ .

Inside a term like $5x$ , the number $5$ is the coefficient: the number multiplied by the variable. The coefficient of $3y$ is $3$ . A variable written with no visible number, such as plain $x$ , has a coefficient of $1$ , because $x$ means $1x$ , one copy of $x$ . Likewise $-x$ means $-1x$ , so its coefficient is $-1$ .

A term that is just a number, with no variable at all, is a constant. In $5x + 3y - 7$ the constant is $-7$ . It is called constant because it never changes: whatever the variables do, that term contributes the same fixed amount.

The parts of 5x + 3y - 7: three terms separated by the + and - signs, two of them with a coefficient on a variable, and one constant term.

Check your understanding

In the expression $9k + 2$ , what is the coefficient of the variable term?

2

k

9

11

Evaluating an expression by substitution

To evaluate an expression you replace each variable with a given number, then do the arithmetic. Replacing a letter with a number is called substitution, and it is the single most useful thing you can do with an expression. Substitution turns an algebraic phrase back into an ordinary calculation.

There is one habit that prevents nearly every mistake. When you substitute, put the number inside parentheses where the variable used to be. The parentheses keep the original operations intact, especially multiplication and signs, so nothing gets accidentally merged or dropped.

Why parentheses matter so much is easiest to see with a coefficient. In $5x$ the variable is being multiplied by $5$ . If $x = 7$ and you simply write the $7$ next to the $5$ , you get the meaningless " $57$ ". Written with parentheses, the same substitution gives $5(7) = 35$ , which is what $5x$ actually means.

Why the same expression gives different values for different inputs#

An expression containing a variable is not a single number; it is a rule that produces a number once the variable is chosen. The rule is fixed, but its output depends on the input.

Take $2x + 1$ . Substituting $x = 3$ replaces the variable by $3$ throughout, giving $2(3) + 1 = 6 + 1 = 7$ . Substituting $x = 5$ instead gives $2(5) + 1 = 10 + 1 = 11$ . The operations, multiply by two and add one, never changed; only the number fed in changed, so the result changed with it.

This is exactly the behaviour the word variable promises. Because the letter can be any number, the expression stands for a family of values at once, one for each choice of the variable. Substitution is how you pick out the single member of that family you currently want. The value you get depends on the input, so changing the input can change the output, as it did here from $7$ to $11$ . Getting a different answer when you change the input is a feature, not an error.

You can carry out a substitution by hand below instead of reading about one. The rectangle takes a width $w$ and a height $h$ , both yours to set, and reports two expressions in those two letters at once. Those two are the area, which is the value of $w \times h$ , and the perimeter, which is the value of $2(w + h)$ . Choosing the two numbers is the substitution; the readout is the evaluation.

Hold the width at $4$ and step the height up from $1$ to $5$ . The area runs $4$ , $8$ , $12$ , $16$ , $20$ , which is the single expression $4h$ evaluated at $h = 1, 2, 3, 4, 5$ : one rule, five inputs, five values. Then set the width to $5$ and the height to $3$ , work out $w \times h$ and $2(w + h)$ on paper, and compare. The figure will not do the substitution for you, but it will tell you whether yours was right.

One rule, many inputs, many values

Width Height

A rectangle 4 units wide and 2 units tall. With w = 4 and h = 2, the expression w times h is 8. The expression 2(w + h) is 12.

A rectangle whose width and height you set. The readout evaluates two expressions in those two letters: the area, w times h, and the perimeter, 2(w + h).

Worked example 3 Evaluate $3x + 4$ when $x = 5$

Substitute $5$ for $x$ , writing it in parentheses so the multiplication stays clear:

3x + 4 = 3(5) + 4.

Now follow the order of operations: multiply before you add.

3(5) + 4 = 15 + 4 = 19.

So when $x = 5$ , the expression $3x + 4$ is worth $19$ .

Worked example 4 Evaluate $2a^2 - b$ when $a = 3$ and $b = 5$

Replace each variable by its value, every one in parentheses:

2a^2 - b = 2(3)^2 - (5).

By the order of operations the exponent goes first, then the multiplication, then the subtraction. Square the $3$ :

2(3)^2 = 2 \cdot 9 = 18.

Then subtract the value of $b$ :

18 - 5 = 13.

So the expression equals $13$ . Notice the exponent applied only to the $3$ , the value of $a$ , because $a^2$ means $a$ is squared, not the whole term.

Substitution works just as smoothly when the value is negative, a fraction, or a decimal. The parentheses matter most in those cases, because they keep the whole value together, sign and all.

Worked example 5 Evaluate $x - 4$ when $x = -2$

Substitute $-2$ for $x$ , keeping it in parentheses so the sign travels with it:

x - 4 = (-2) - 4.

Subtracting $4$ from $-2$ moves four further down the number line, into the negatives:

(-2) - 4 = -6.

So when $x = -2$ , the expression $x - 4$ is worth $-6$ . Writing the input as $(-2)$ rather than a bare $-2$ is what keeps the leading minus sign from being misread.

Check your understanding

Evaluate $4n - 3$ when $n = 2$ .

5

8

3

11

Common mistakes

Reversing “less than” and “subtracted from.” “Five less than $x$ ” is $x - 5$ , not $5 - x$ . Start from the quantity named, then take the amount away. The same flip catches “fewer than” and “subtracted from.”
Reading $3n$ as a two-digit number. Side-by-side notation means multiply, but only between a number and a variable: $3n = 3 \times n$ . The plain digits $34$ still mean thirty-four.
Treating an expression like an equation. An expression has no equals sign, so you evaluate it; you do not “solve” it. Only an equation, with its equals sign, gets solved.
Forgetting the invisible coefficient $1$ . Plain $x$ means $1x$ and $-x$ means $-1x$ . The coefficient is $1$ or $-1$ , not zero and not missing.
Dropping parentheses when substituting. Replace $x$ with $(7)$ , not a bare $7$ , so $5x$ becomes $5(7) = 35$ rather than the nonsense $57$ , and a negative input keeps its sign. This single habit removes most substitution errors.
Multiplying before grouping in “twice the sum.” “Twice the sum of $x$ and $4$ ” is $2(x + 4)$ , not $2x + 4$ . Parentheses must hold the whole sum together before it is doubled.

Key takeaways

A variable is a letter standing for a number that is unknown or free to change. The same letter can take different values on different occasions.
An expression is a phrase ( $3n$ , $x + 5$ ) that names a value. An equation joins two expressions with an equals sign ( $3n = 12$ ) and can be true or false. You evaluate expressions and solve equations.
Translate words by naming the unknown with a letter and converting each phrase to an operation. Mind the reversal in “less than,” and use parentheses for “twice the sum” and similar groupings.
The parts of an expression are its terms (separated by $+$ and $-$ ), its coefficient, and its constant (a number term). A coefficient is the number multiplying a variable, with plain $x$ meaning $1x$ .
To evaluate, substitute each variable’s value in parentheses and follow the order of operations: $3x + 4$ at $x = 5$ is $3(5) + 4 = 19$ .
A number written next to a variable means multiply: $3n$ is $3 \times n$ .

Variables and Expressions

Learning goals

What a variable is

Expressions versus equations

Translating words into expressions

The parts of an expression

Evaluating an expression by substitution

Common mistakes

Practice

Multiple Choice Questions (MCQ)

Free Response Questions (FRQ)