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 names a single value, once you know its variables. It can be a number or a variable on its own, or several of those combined with operations ( $+$ , $-$ , $\times$ , $\div$ , exponents). It is a phrase, the algebraic version of a number. Each of these is an expression:

7, \qquad x, \qquad 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. Whether that statement is true can depend on which number the variable turns out to be. 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

Check your understanding

The expression $2n + 5$ and the equation $2n + 5 = 11$ share the same left side. What do you do with each one?

A Evaluate the expression by substituting a value for

n

; solve the equation to find the value of

n

that makes it true. B Solve both, since both contain the letter

n

. C Evaluate both, since neither has a definite value until

n

is chosen. D Solve the expression to find

n

, then evaluate the equation with that value.

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. Start by naming the unknown with a letter. Then read the whole phrase for its meaning: what quantity are you starting from, what happens to it, and in what order? Once you can answer that, this vocabulary helps you write down each operation:

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 it was the meaning of the phrase, not just its words, that decided the operation and the order.

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 make sure “twice the sum” doubles the whole sum, not just part of it.

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 a $+$ or $-$ sign written at the main level of the expression, not one tucked inside parentheses; a grouped piece like $(x + 4)$ from the last example stays together as one piece until the parentheses are removed. By convention a term carries the sign written in front of it, so the subtracted piece here 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

Which statement about the expression $9k + 2$ is correct?

A It has two terms,

9k

and

2

; the coefficient of

9k

9

, and the constant is

2

. B It has one term,

9k + 2

, with coefficient

11

. C It has two terms,

9k

and

2

; the coefficient of

9k

2

, and the constant is

9

. D It has two terms,

9

and

k + 2

; the coefficient of

9

k

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 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 " $57$ ", the number fifty-seven, which is the wrong number entirely, not the product $5 \times 7$ . Written with parentheses, the same substitution gives $5(7) = 35$ , which is what $5x$ actually means.

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 $19$ .

Why the same expression gives different values for different inputs#

An expression with a variable is not one fixed number. It is a set of instructions that produces a number once you choose the variable’s value; the instructions stay the same, but what you choose does not.

Take $2x + 1$ . Substituting $x = 3$ gives $2(3) + 1 = 6 + 1 = 7$ . Substituting $x = 5$ instead gives $2(5) + 1 = 10 + 1 = 11$ . The instructions, multiply by two and add one, are identical both times. Only the number substituted for $x$ changed, from $3$ to $5$ , and that alone is what changed the result, from $7$ to $11$ .

This is exactly what the word variable promises: because $x$ can be any number, the expression can produce a different value for each number you choose. Getting a different answer when you substitute a different value is expected, not a mistake.

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 evaluated at those values at once: the area, $w \times h$ , and the perimeter, $2(w + h)$ . Setting the two numbers is the substitution; the readout is the evaluation.

Try it first with one letter changing. Set $w = 4$ , then step $h$ from $1$ to $5$ and watch the area. It runs $4$ , $8$ , $12$ , $16$ , $20$ , the single expression $4h$ evaluated at five different heights: one expression, five inputs, five results. Then set $w = 5$ and $h = 3$ . Before you look at the readout, work out $w \times h$ and $2(w + h)$ yourself on paper, and check your answers against what the rectangle reports.

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 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. The parentheses matter most in that case, because they keep the whole value, sign included, together as one piece.

Worked example 5 Evaluate $3x + 1$ when $x = -2$

Substitute $-2$ for $x$ , keeping it in parentheses so the multiplication and the sign both stay clear:

3x + 1 = 3(-2) + 1.

The parentheses matter here more than ever. Without them, " $3-2$ " would read as a subtraction instead of the multiplication $3 \times (-2)$ that $3x$ actually means, and the sign on the $2$ could be lost along the way.

Now follow the order of operations, multiplying before adding:

3(-2) + 1 = -6 + 1 = -5.

So when $x = -2$ , the expression $3x + 1$ is worth $-5$ .

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 wrong number $57$ , 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. A number written right beside a variable means multiply, so $3n$ is $3 \times n$ .
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 a $+$ or $-$ at the main level), 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 $3(5) + 4 = 19$ .

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)

Core practice