How do I know when to use parentheses in an expression?

Use parentheses when you need to group operations that should happen together before other operations. If a phrase uses a word like "sum of" or "product of" followed by multiple things, those things need to be grouped. For example, "the product of 3 and the sum of x and 2" is 3(x + 2), with parentheses around x + 2 so they're added before multiplying by 3. Without parentheses, 3x + 2 means multiply x by 3 first, then add 2—a different result.

What's the difference between writing x + 5 and 5 + x?

Mathematically, they're equal because addition is commutative—the order doesn't matter. Both equal the same value. However, when translating from words, the phrase "5 more than a number" naturally writes as x + 5 (a number, then add 5), while "a number added to 5" could write as 5 + x. Either form is correct, but the first usually matches the word order better. With subtraction, order matters: x - 5 and 5 - x give different values.

Why do I need to write parentheses when I substitute, like 2(3) instead of 23?

Writing 2(3) makes it clear that you're multiplying 2 by 3. If you write 23, it looks like the number twenty-three, which is completely different. This is especially important for negative numbers: 2(-3) is obviously -6, but 2-3 looks like a subtraction problem. Parentheses prevent confusion and help you avoid calculation mistakes.

When I evaluate a formula, does it matter which variable I substitute first?

No, it doesn't matter. In a formula like A = 1/2bh, you can substitute b first or h first—you'll get the same answer either way. Just make sure you substitute all the variables and follow the order of operations when you calculate. It's a good idea to write out your substitution clearly so you can check your work.

M6MATH-7.2

Writing & Evaluating Algebraic Expressions

Learn to translate word phrases into algebraic expressions and evaluate them by substituting variable values. Master formulas and real-world problems.

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What this lesson covers

Algebra lets you describe situations using letters and numbers instead of always working with specific numbers. In this lesson, you'll learn how to take everyday phrases like "three more than a number" and write them as algebraic expressions, then find their value when you know what the number is. This skill is the foundation for solving real problems in science, business, and engineering.

Translating Words to Algebraic Expressions

When you convert a word phrase into an algebraic expression, you're learning to think like an algebraist. Start by identifying what the unknown quantity is—that becomes your variable, usually

x

n

, or

y

. Then identify the operation the phrase describes.

Some common phrases and what they mean:

Phrase	Operation	Expression
a number plus 5	addition	$x + 5$
3 less than a number	subtraction	$x - 3$
twice a number	multiplication	$2x$
a number divided by 4	division	$\frac{x}{4}$
the sum of a number and 7	addition	$x + 7$
the product of 6 and a number	multiplication	$6x$
a number squared	exponent	$x^2$

Key insight: "Less than" reverses the order. "5 less than a number" is

x - 5

, NOT

5 - x

. The number comes first. Similarly, "divided by" reverses order: "10 divided by a number" is

\frac{10}{x}

, but "a number divided by 10" is

\frac{x}{10}

. Pay close attention to phrase order to get the expression right.

Building More Complex Expressions

Once you're comfortable with simple operations, you can handle phrases that combine multiple operations. Break the phrase into smaller parts and build up.

Example: "twice the sum of a number and 3". Don't jump straight to writing

2x + 3

—that's wrong because it means "twice the number, plus 3." Instead:

Identify the core: the sum of a number and 3 =

x + 3

. Now apply "twice" (multiply by 2):

2(x + 3)

2 \times (x + 3)

.

Another example: "five more than the product of 4 and a number". The product of 4 and a number is

4x

. Five more than that is

4x + 5

.

The order of operations matters when you write and when you evaluate. Parentheses group operations that must happen first. Without parentheses, multiplication and division happen before addition and subtraction. A common mistake is forgetting parentheses when the phrase describes something that should be grouped—like "the sum of a number and 3" being multiplied.

Evaluating Expressions

Once you have an expression, evaluating it means substituting a number for the variable and calculating the result. This tells you the value of the expression for that particular input.

To evaluate

3x + 5

when

x = 2

:

Substitute 2 for every

x

3(2) + 5

. Then calculate using the order of operations. Multiply first:

6 + 5 = 11

.

To evaluate

2(x + 3)

when

x = 4

:

Substitute 4 for

x

2(4 + 3)

. Do what's in parentheses first:

2(7) = 14

.

When you substitute, write the number in parentheses to keep your work clear and avoid errors. For example, write

3(2) + 5

, not

32 + 5

. This is especially important with negative numbers:

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

is much clearer than

3-2 + 5

, which looks confusing.

A formula is just a special algebraic expression that describes a real-world relationship. The perimeter of a rectangle is

P = 2l + 2w

where

l

is length and

w

is width. The area of a triangle is

A = \frac{1}{2}bh

where

b

is base and

h

is height. Evaluating a formula works exactly the same way: substitute the known values and calculate.

Common Mistakes and How to Avoid Them

Students often reverse operations when translating phrases. Remember: "less than" puts the subtraction in reverse order. "8 less than

x

" is

x - 8

, not

8 - x

. The phrase "a number less 8" is also

x - 8

. Think about it: if I start with 10 and want "8 less than 10," I get

10 - 8 = 2

. That's correct order.

Another common error: forgetting to use parentheses. "The sum of a number and 5, multiplied by 2" is

2(x + 5)

, not

2x + 5

. These give different answers when you evaluate them.

When evaluating, remember that

2x

means

2 \times x

. If

x = 3

, then

2x = 6

, not

23

. Write

2(3)

to make it obvious.

With formulas, use the variables correctly. In

A = \frac{1}{2}bh

, the

b

and

h

are different quantities (base and height), so don't mix them up. Label what each number represents before you substitute.

Key terms

Algebraic expression.: A mathematical phrase using variables, numbers, and operation symbols—for example, $3x + 5$ or $2(x + 3)$ .
Variable.: A letter or symbol that stands for an unknown number or a number that can change. Common variables are $x$ , $y$ , or $n$ .
Evaluate.: To find the value of an expression by substituting known numbers for the variables and calculating the result.
Substitute.: To replace a variable with a specific number value.
Coefficient.: The number multiplied by a variable. In the expression $5x$ , the coefficient is 5.
Formula.: An algebraic equation that describes a relationship between quantities in a real-world context—for example, $A = lw$ for the area of a rectangle.
Order of operations.: The rules for which calculations to perform first: parentheses, exponents, multiplication and division (left to right), then addition and subtraction (left to right). Abbreviated as PEMDAS or BODMAS.

Worked example

The drama club is selling tickets to a play. Student tickets cost 8 dollars and adult tickets cost 12 dollars. Write an algebraic expression for the total revenue if they sell

s

student tickets and

a

adult tickets. Then evaluate the expression if they sell 45 student tickets and 30 adult tickets.

Step 1: Identify what you're finding. Total revenue means the total amount of money from all tickets sold.

Step 2: Set up the expression for each type of ticket. Student tickets cost 8 dollars each, and they sell

s

of them, so that's

8s

dollars. Adult tickets cost 12 dollars each, and they sell

a

of them, so that's

12a

dollars.

Step 3: Combine the parts. Total revenue =

8s + 12a

.

Step 4: Evaluate when

s = 45

and

a = 30

. Substitute both values:

8(45) + 12(30)

.

Step 5: Calculate. Multiply first:

360 + 360 = 720

dollars.

The expression is

8s + 12a

, and when they sell 45 student tickets and 30 adult tickets, the total revenue is 720 dollars.

Practice questions

Write an algebraic expression for "7 less than twice a number." Then evaluate it when the number is 6.

Answer: $2x - 7$ ; the value is 5

The phrase breaks into two parts: "twice a number" is

2x

. Then "7 less than" that means we subtract 7. The order matters—don't write

7 - 2x

; that's backwards. So the expression is

2x - 7

. When

x = 6

2(6) - 7 = 12 - 7 = 5

. A common wrong answer is

2x + 7 = 19

, which comes from reading "less than" as addition instead of subtraction.

The cost to rent a bike is 5 dollars plus 2 dollars per hour. Which expression represents the total cost for

h

hours?

$5 + 2h$
$5h + 2$
$7h$
$(5 + 2)h$

Answer: $5 + 2h$

The 5 dollars is a flat fee—it happens once, no matter how many hours. The 2 dollars per hour means 2 dollars multiplied by the number of hours, which is

2h

. So the total is

5 + 2h

. The choice

5h + 2

is wrong because it multiplies the initial fee by hours. The choice

7h

is wrong because it ignores that 5 dollars is a one-time cost. The choice

(5 + 2)h

is wrong because it would mean the entire 7 dollars is charged per hour.

The area of a trapezoid is given by the formula

A = \frac{1}{2}(b_1 + b_2)h

, where

b_1

and

b_2

are the lengths of the two parallel sides and

h

is the height. Find the area of a trapezoid with

b_1 = 6

cm,

b_2 = 10

cm, and

h = 4

cm.

Answer: 32 square cm

Substitute each value into the formula:

A = \frac{1}{2}(6 + 10)(4)

. Work inside the parentheses first:

6 + 10 = 16

. Then

A = \frac{1}{2}(16)(4)

. Multiply left to right:

\frac{1}{2} \times 16 = 8

, then

8 \times 4 = 32

. So the area is 32 square cm. A common error is forgetting to add the two bases first—if you multiply first, you'd get the wrong answer.

FAQ

How do I know when to use parentheses in an expression?: Use parentheses when you need to group operations that should happen together before other operations. If a phrase uses a word like "sum of" or "product of" followed by multiple things, those things need to be grouped. For example, "the product of 3 and the sum of $x$ and 2" is $3(x + 2)$ , with parentheses around $x + 2$ so they're added before multiplying by 3. Without parentheses, $3x + 2$ means multiply $x$ by 3 first, then add 2—a different result.
What's the difference between writing $x + 5$ and $5 + x$ ?: Mathematically, they're equal because addition is commutative—the order doesn't matter. Both equal the same value. However, when translating from words, the phrase "5 more than a number" naturally writes as $x + 5$ (a number, then add 5), while "a number added to 5" could write as $5 + x$ . Either form is correct, but the first usually matches the word order better. With subtraction, order matters: $x - 5$ and $5 - x$ give different values.
Why do I need to write parentheses when I substitute, like $2(3)$ instead of $23$ ?: Writing $2(3)$ makes it clear that you're multiplying 2 by 3. If you write $23$ , it looks like the number twenty-three, which is completely different. This is especially important for negative numbers: $2(-3)$ is obviously $-6$ , but $2-3$ looks like a subtraction problem. Parentheses prevent confusion and help you avoid calculation mistakes.
When I evaluate a formula, does it matter which variable I substitute first?: No, it doesn't matter. In a formula like $A = \frac{1}{2}bh$ , you can substitute $b$ first or $h$ first—you'll get the same answer either way. Just make sure you substitute all the variables and follow the order of operations when you calculate. It's a good idea to write out your substitution clearly so you can check your work.

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