Solved Exercises on Writing Algebraic Expressions in Grade 7

Master writing algebraic expressions: translating words to math, operations, variables, and real-world applications through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Addition Expressions
Exercise 1
Write an algebraic expression for: "the sum of a number and 12"
Definition:

Algebraic expression: A mathematical phrase that uses numbers, variables, and operation symbols to represent relationships.

Translation method:
  1. Identify the unknown quantity (variable)
  2. Identify the operation(s)
  3. Translate words to mathematical symbols
  4. Write the complete expression
Word Phrase
"the sum of a number and 12"
Identify Variable
Let x = the unknown number
Algebraic Expression
x + 12
Step 1: Identify the unknown

"a number" means we need a variable. Let x = the unknown number

Step 2: Identify the operation

"sum" means addition (+)

Step 3: Identify the known value

"and 12" means we add 12

Step 4: Write the expression

x + 12

x + 12
Final answer:

The algebraic expression is: x + 12

Applied rules:

• Variable identification: Words like "a number", "some number", or "unknown" indicate variables

• Operation words: "sum", "more than", "increased by" mean addition

• Order matters: "12 more than a number" is x + 12, not 12 + x

2 Multiplication Expressions
Exercise 2
Write an algebraic expression for: "seven times a number"
Definition:

Multiplication in algebra: Can be represented as 7x, 7·x, or 7(x). The multiplication symbol is often omitted between a number and variable.

Word Phrase
"seven times a number"
Identify Variable
Let n = the unknown number
Algebraic Expression
7n
Step 1: Identify the unknown

"a number" means we need a variable. Let n = the unknown number

Step 2: Identify the multiplier

"seven" means 7

Step 3: Identify the operation

"times" means multiplication (×)

Step 4: Write the expression

7n (multiplication symbol is usually omitted)

7n
Final answer:

The algebraic expression is: 7n

Applied rules:

• Multiplication words: "times", "product of", "multiplied by" mean multiplication

• Algebraic notation: 7n is preferred over 7 × n or 7·n

• Number first: Write the constant before the variable (7n, not n7)

3 Division Expressions
Exercise 3
Write an algebraic expression for: "the quotient of a number and 8"
Definition:

Division in algebra: Can be represented as \(\frac{x}{8}\) or x ÷ 8. Fraction form is preferred.

Word Phrase
"the quotient of a number and 8"
Identify Variable
Let y = the unknown number
Algebraic Expression
\(\frac{y}{8}\)
Step 1: Identify the unknown

"a number" means we need a variable. Let y = the unknown number

Step 2: Identify the operation

"quotient" means division (÷)

Step 3: Identify dividend and divisor

"a number" is divided by "8", so y ÷ 8

Step 4: Write in fraction form

\(\frac{y}{8}\) (preferred notation)

\(\frac{y}{8}\)
Final answer:

The algebraic expression is: \(\frac{y}{8}\)

Applied rules:

• Division words: "quotient", "divided by", "ratio of" mean division

• Fraction form: \(\frac{x}{n}\) is preferred over x ÷ n

• Order matters: "a number divided by 8" is \(\frac{x}{8}\), not \(\frac{8}{x}\)

Algebraic Expressions Keywords and Rules
x + 5
Basic Expression
Addition
+, increased by, more than, sum of, total
Examples: x + 3, 5 + y
Subtraction
-, decreased by, less than, difference, minus
Examples: x - 7, 10 - y
Multiplication
×
Examples: 3x, x(5), 4(y)
Key definitions:

Variable: A symbol (usually a letter) that represents an unknown value.

Constant: A fixed value that does not change.

Coefficient: The numerical factor in a term containing a variable.

Term: A number, variable, or product of numbers and variables.

Algebraic expression: A combination of terms connected by operation symbols.

Translation methodology:
  1. Read carefully: Identify what the unknown represents
  2. Look for keywords: Identify operation words
  3. Assign variables: Choose appropriate letters
  4. Build expression: Connect terms with operations
  5. Check order: Ensure correct sequence of operations
Tip 1: "More than" and "less than" reverse the order: "5 more than x" is x + 5.
Tip 2: Use different letters for different unknowns to avoid confusion.
Tip 3: Parentheses help clarify grouping when writing complex expressions.
Tip 4: Read the expression back in words to verify it matches the original phrase.
Common errors: Reversing the order of subtraction, forgetting to use variables for unknowns, misinterpreting "more than" and "less than" phrases, confusing addition with multiplication keywords.
Exam preparation: Memorize operation keywords, practice translating word phrases, understand the importance of order in operations, work with multiple variables.
Essential translation rules:

• Addition: sum, plus, increased by, more than, total

• Subtraction: difference, minus, decreased by, less than, subtracted from

• Multiplication: product, times, multiplied by, of, twice

• Division: quotient, divided by, ratio, per, out of

• Powers: squared, cubed, to the power of

Solution: Exercises 4 to 5
4 Complex Expressions
Exercise 4
Write an algebraic expression for: "five less than three times a number"
Definition:

Complex expressions: Contain multiple operations that must be performed in the correct order. Follow the order of operations when translating.

Word Phrase
"five less than three times a number"
Identify Operations
Multiplication and subtraction
Algebraic Expression
3x - 5
Step 1: Identify the unknown

"a number" means we need a variable. Let x = the unknown number

Step 2: Break down the phrase

"three times a number" = 3x

"five less than" means subtract 5 from the previous result

Step 3: Apply the operations

"Five less than three times a number" = 3x - 5

Step 4: Verify the expression

If x = 10: 3(10) - 5 = 30 - 5 = 25

"Five less than three times ten" = 30 - 5 = 25 ✓

3x - 5
Final answer:

The algebraic expression is: 3x - 5

Applied rules:

• Order matters: "less than" reverses the order: "5 less than x" is x - 5

• Complex phrases: Break down into simpler parts first

• Verification: Test with actual numbers to confirm accuracy

5 Real-World Application
Exercise 5
Sarah earns $15 per hour and works h hours. She also receives a $50 bonus. Write an expression for her total earnings.
Definition:

Real-world applications: Translate practical situations into mathematical expressions to model and solve problems.

Components
Hourly wage × hours + bonus
Identify Variables
h = hours worked
Algebraic Expression
15h + 50
Step 1: Identify what we're finding

Total earnings = hourly pay + bonus

Step 2: Identify known values

Hourly rate = $15, Bonus = $50

Step 3: Identify unknowns

Hours worked = h (variable)

Step 4: Write the expression

Total earnings = (hourly rate × hours) + bonus

Total earnings = 15h + 50

Step 5: Verify with example

If h = 20: 15(20) + 50 = 300 + 50 = $350

This makes sense: $15/hour × 20 hours + $50 bonus = $350

15h + 50
Final answer:

The algebraic expression for Sarah's total earnings is: 15h + 50

Applied rules:

• Real-world modeling: Identify the relationship between quantities

• Define variables clearly: State what each variable represents

• Verification: Test the expression with real values to ensure accuracy

Detailed Algebraic Expressions Guide
ax + b
Linear Expression
Key definitions:

Algebraic expression: A mathematical phrase that combines numbers, variables, and operation symbols to represent relationships and quantities.

Variable: A symbol (usually a letter) that represents an unknown or changing value.

Constant: A fixed value that does not change throughout the problem.

Coefficient: The numerical factor that multiplies a variable in a term.

Term: A single number, variable, or the product of numbers and variables.

Complete translation methodology:
  1. Read the entire phrase: Understand the full context
  2. Identify unknowns: Assign variables to unknown quantities
  3. Find operation words: Match keywords to mathematical operations
  4. Build the expression: Construct using proper mathematical notation
  5. Verify correctness: Check that the expression matches the original meaning
Tip 1: Phrases like "more than" and "less than" reverse the order: "5 more than x" is x + 5.
Tip 2: Use different variables for different unknowns (x for first, y for second, etc.).
Tip 3: Parentheses can help organize complex expressions: "the sum of x and 3, multiplied by 2" is (x + 3) × 2.
Tip 4: Always read your final expression back in words to verify it matches the original phrase.
Common errors: Misreading "less than" and "more than" phrases, reversing subtraction order incorrectly, using wrong operation words, forgetting to include all parts of the expression.
Applications: Modeling real-world situations, solving equations, representing relationships, calculating costs, measuring distances, analyzing patterns, and preparing for higher-level mathematics.
Essential translation keywords:

• Addition: sum, plus, increased by, more than, total, combined, added to

• Subtraction: difference, minus, decreased by, less than, subtracted from, reduced by

• Multiplication: product, times, multiplied by, of, twice, double, triple

• Division: quotient, divided by, ratio, per, out of, fraction of

• Powers: squared, cubed, to the power of, raised to

Algebraic Expression Translation Guide

📊
Addition Words

sum of x and 5 → x + 5

3 more than y → y + 3

increased by 7 → x + 7

Subtraction Words

difference of x and 5 → x - 5

3 less than y → y - 3

decreased by 7 → x - 7

Multiplication Words

product of x and 5 → 5x

twice a number → 2x

triple y → 3y

Division Words

quotient of x and 5 → x/5

ratio of x to 3 → x/3

x divided by 4 → x/4

Questions & Answers

Question: I'm confused about "5 less than x" versus "x less than 5". Aren't they the same thing?

Answer: No, these are completely different! The word "than" signals that the order is reversed:

"5 less than x":

  • Start with x
  • Subtract 5 from it
  • Expression: x - 5

"x less than 5":

  • Start with 5
  • Subtract x from it
  • Expression: 5 - x

Example: If x = 8:

  • "5 less than x" = 8 - 5 = 3
  • "x less than 5" = 5 - 8 = -3

Remember: "less than" and "more than" always reverse the order!

Question: Why do we use letters like x, y, z for variables? Can't we just use any letter?

Answer: Yes, you can use any letter as a variable! The choice of x, y, z is just a convention:

Traditional choices:

  • x, y, z are commonly used for unknowns
  • a, b, c are often used for constants
  • n is frequently used for numbers
  • t is often used for time

Best practices:

  • Choose meaningful letters when possible (h for hours, d for distance)
  • Use different letters for different unknowns
  • Avoid letters that might be confused with numbers (avoid 'l' which looks like '1')

Example: If writing about age, you could use 'a', 'age', or any letter - it's your choice!

Question: When do I need parentheses in algebraic expressions? For example, when should I write (x + 3) × 2 versus x + 3 × 2?

Answer: Parentheses are crucial for indicating order of operations! Without them, multiplication happens before addition:

Without parentheses: x + 3 × 2

  • Multiply first: 3 × 2 = 6
  • Then add: x + 6

With parentheses: (x + 3) × 2

  • Add first: x + 3
  • Then multiply by 2: 2(x + 3) = 2x + 6

When to use parentheses:

  • When you want addition/subtraction to happen before multiplication/division
  • To group terms that belong together
  • To clarify the intended order of operations
  • When translating phrases like "the sum of x and 3, multiplied by 2"