Solved Exercises on Simplifying Expressions in Grade 7

Master simplifying expressions: combining like terms, distributive property, factoring, and multi-step simplification through these 10 detailed exercises with visual learning tools.

Solution: Exercises 1 to 3
1 Combining Like Terms
Exercise 1
Simplify: 3x + 5y - 2x + 4y + 7
Definition:

Like terms: Terms that have the same variable parts raised to the same powers.

Method:
  1. Group terms with the same variables together
  2. Add or subtract the coefficients of like terms
  3. Combine constants separately
  4. Write the simplified expression
Step 1: Identify like terms

Like terms: 3x and -2x (both have variable x), 5y and 4y (both have variable y)

Step 2: Group like terms

(3x - 2x) + (5y + 4y) + 7

Step 3: Combine like terms

(3 - 2)x + (5 + 4)y + 7 = 1x + 9y + 7

Step 4: Write final expression

x + 9y + 7

x + 9y + 7
Final answer:

3x + 5y - 2x + 4y + 7 simplifies to x + 9y + 7.

Applied rules:

• Combining like terms: Add coefficients of like terms

• Variable preservation: Keep variables unchanged when combining

• Constant terms: Combine numbers separately

2 Distributive Property
Exercise 2
Simplify: 4(2x - 3) + 5(x + 1)
Definition:

Distributive property: a(b + c) = ab + ac

Step 1: Apply distributive property to first term

4(2x - 3) = 4 × 2x + 4 × (-3) = 8x - 12

Step 2: Apply distributive property to second term

5(x + 1) = 5 × x + 5 × 1 = 5x + 5

Step 3: Combine results

8x - 12 + 5x + 5

Step 4: Combine like terms

(8x + 5x) + (-12 + 5) = 13x - 7

4(2x - 3)
= 8x - 12
5(x + 1)
= 5x + 5
Combined
= 13x - 7
13x - 7
Final answer:

4(2x - 3) + 5(x + 1) simplifies to 13x - 7.

Applied rules:

• Distributive property: Multiply each term inside parentheses

• Sign handling: Distribute negative signs correctly

• Like terms: Combine after distributing

3 Factoring Out Common Factors
Exercise 3
Factor: 12x + 18
Definition:

Factoring: The process of writing an expression as a product of factors.

Step 1: Find the GCF of coefficients

Factors of 12: 1, 2, 3, 4, 6, 12

Factors of 18: 1, 2, 3, 6, 9, 18

GCF = 6

Step 2: Factor out the GCF

12x + 18 = 6(2x) + 6(3) = 6(2x + 3)

Step 3: Verify the factorization

6(2x + 3) = 6 × 2x + 6 × 3 = 12x + 18 ✓

Original: 12x + 18
= 6(2x + 3)
6(2x + 3)
Final answer:

12x + 18 factors to 6(2x + 3).

Applied rules:

• Greatest common factor: Find largest factor that divides all terms

• Distributive property: Reverse to factor out common terms

• Verification: Multiply back to check correctness

Solution: Exercises 4 to 6
4 Multiple Variables
Exercise 4
Simplify: 5x + 3y - 2x + 4y - x + 2y
Definition:

Multiple variables: Expressions containing more than one variable.

Step 1: Group like terms

Terms with x: 5x, -2x, -x

Terms with y: 3y, 4y, 2y

Step 2: Combine x terms

5x - 2x - x = (5 - 2 - 1)x = 2x

Step 3: Combine y terms

3y + 4y + 2y = (3 + 4 + 2)y = 9y

Step 4: Write the simplified expression

2x + 9y

x terms: 5x - 2x - x
= 2x
y terms: 3y + 4y + 2y
= 9y
2x + 9y
Final answer:

5x + 3y - 2x + 4y - x + 2y simplifies to 2x + 9y.

Applied rules:

• Group by variable: Combine terms with the same variable separately

• Coefficient addition: Add numerical coefficients only

• Variable preservation: Keep variables unchanged

5 Complex Distribution
Exercise 5
Simplify: 3(4x - 2) - 2(3x + 5)
Definition:

Distribution with subtraction: Apply the distributive property to each term.

Step 1: Distribute 3 to first parentheses

3(4x - 2) = 3 × 4x + 3 × (-2) = 12x - 6

Step 2: Distribute -2 to second parentheses

-2(3x + 5) = -2 × 3x + (-2) × 5 = -6x - 10

Step 3: Combine results

12x - 6 - 6x - 10

Step 4: Combine like terms

(12x - 6x) + (-6 - 10) = 6x - 16

3(4x - 2)
= 12x - 6
-2(3x + 5)
= -6x - 10
Combined
= 6x - 16
6x - 16
Final answer:

3(4x - 2) - 2(3x + 5) simplifies to 6x - 16.

Applied rules:

• Distributive property: Multiply each term inside parentheses

• Sign distribution: Distribute negative signs to all terms

• Like terms: Combine after distribution

6 Mixed Operations
Exercise 6
Simplify: 2x + 3(4 - x) - 5 + 2x
Definition:

Mixed operations: Expressions requiring multiple simplification techniques.

Step 1: Distribute 3 to the parentheses

3(4 - x) = 3 × 4 + 3 × (-x) = 12 - 3x

Step 2: Rewrite the expression

2x + 12 - 3x - 5 + 2x

Step 3: Group like terms

(2x - 3x + 2x) + (12 - 5)

Step 4: Combine like terms

(2 - 3 + 2)x + 7 = 1x + 7 = x + 7

Distribute: 3(4 - x)
= 12 - 3x
Rewrite: 2x + 12 - 3x - 5 + 2x
= x + 7
x + 7
Final answer:

2x + 3(4 - x) - 5 + 2x simplifies to x + 7.

Applied rules:

• Distribution first: Perform distribution before combining like terms

• Order of operations: Address parentheses first

• Like terms: Combine after distribution is complete

Expression Simplification Visual Guide
a(b+c) = ab + ac
Distributive Property
Like Terms
Same variable parts
Distribution
Multiply each term
Factoring
Extract common factors
Simplification
Reduce to simplest form
Simplification Process:
Step 1: Identify and group like terms
Step 2: Apply distributive property if needed
Step 3: Combine like terms by adding coefficients
Step 4: Factor out common terms if possible
Step 5: Verify the simplified expression
Tip 1: Always check that variables match exactly before combining terms.
Tip 2: When distributing, multiply by every term inside the parentheses.
Tip 3: Factor out the greatest common factor for complete simplification.
Common errors: Forgetting to distribute negative signs, combining unlike terms, miscalculating coefficients.
Success strategies: Systematic approach, verification, careful sign handling.
Essential concepts:

• Like terms: Same variable parts with same exponents

• Distribution: Multiply outside factor by each term inside

• Factoring: Reverse of distribution

• Simplification: Express in most compact form

Solution: Exercises 7 to 10
7 Complex Expression
Exercise 7
Simplify: 2(3x + 4) - 3(2x - 1) + 5x - 7
Definition:

Complex expression: An expression requiring multiple simplification steps.

Step 1: Distribute 2 to first parentheses

2(3x + 4) = 6x + 8

Step 2: Distribute -3 to second parentheses

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

Step 3: Rewrite the expression

6x + 8 - 6x + 3 + 5x - 7

Step 4: Group like terms

(6x - 6x + 5x) + (8 + 3 - 7)

Step 5: Combine like terms

(6 - 6 + 5)x + (11 - 7) = 5x + 4

Distribute: 2(3x + 4)
= 6x + 8
Distribute: -3(2x - 1)
= -6x + 3
Combine: (6x - 6x + 5x) + (8 + 3 - 7)
= 5x + 4
5x + 4
Final answer:

2(3x + 4) - 3(2x - 1) + 5x - 7 simplifies to 5x + 4.

Applied rules:

• Distribution: Apply to each term in parentheses

• Sign handling: Be careful with negative distribution

• Like terms: Combine coefficients of like terms

8 Multi-step Problem
Exercise 8
Simplify: 4x - 2(3x + 5) + 3(2 - x) + 6x
Definition:

Multi-step simplification: An expression requiring sequential operations.

Step 1: Distribute -2 to first parentheses

-2(3x + 5) = -6x - 10

Step 2: Distribute 3 to second parentheses

3(2 - x) = 6 - 3x

Step 3: Rewrite the expression

4x - 6x - 10 + 6 - 3x + 6x

Step 4: Group like terms

(4x - 6x - 3x + 6x) + (-10 + 6)

Step 5: Combine like terms

(4 - 6 - 3 + 6)x + (-4) = 1x - 4 = x - 4

Distribute: -2(3x + 5)
= -6x - 10
Distribute: 3(2 - x)
= 6 - 3x
Combine: (4x - 6x - 3x + 6x) + (-10 + 6)
= x - 4
x - 4
Final answer:

4x - 2(3x + 5) + 3(2 - x) + 6x simplifies to x - 4.

Applied rules:

• Sequential distribution: Apply distribution to each parentheses separately

• Sign tracking: Keep track of positive and negative signs

• Like terms: Group and combine like terms at the end

9 Factoring Challenge
Exercise 9
Factor completely: 15x² + 10x
Definition:

Complete factoring: Factor out the greatest common factor from all terms.

Step 1: Find GCF of coefficients

GCF of 15 and 10 is 5

Step 2: Find GCF of variables

GCF of x² and x is x

Step 3: Factor out GCF

15x² + 10x = 5x(3x) + 5x(2) = 5x(3x + 2)

Step 4: Verify the factorization

5x(3x + 2) = 5x × 3x + 5x × 2 = 15x² + 10x ✓

Original: 15x² + 10x
= 5x(3x + 2)
5x(3x + 2)
Final answer:

15x² + 10x factors to 5x(3x + 2).

Applied rules:

• GCF identification: Find greatest factor common to all terms

• Variable factoring: Take lowest power of each variable

• Verification: Multiply back to confirm factorization

10 Advanced Simplification
Exercise 10
Simplify: 2(4x - 3y) - 3(2x + y) + 4(x - 2y)
Definition:

Advanced simplification: An expression with multiple variables requiring careful distribution.

Step 1: Distribute 2 to first parentheses

2(4x - 3y) = 8x - 6y

Step 2: Distribute -3 to second parentheses

-3(2x + y) = -6x - 3y

Step 3: Distribute 4 to third parentheses

4(x - 2y) = 4x - 8y

Step 4: Combine all terms

8x - 6y - 6x - 3y + 4x - 8y

Step 5: Group like terms

(8x - 6x + 4x) + (-6y - 3y - 8y)

Step 6: Combine like terms

(8 - 6 + 4)x + (-6 - 3 - 8)y = 6x - 17y

Distribute: 2(4x - 3y)
= 8x - 6y
Distribute: -3(2x + y)
= -6x - 3y
Distribute: 4(x - 2y)
= 4x - 8y
Combine: (8x - 6x + 4x) + (-6y - 3y - 8y)
= 6x - 17y
6x - 17y
Final answer:

2(4x - 3y) - 3(2x + y) + 4(x - 2y) simplifies to 6x - 17y.

Applied rules:

• Distribution: Apply to each term in each parentheses

• Multi-variable: Group terms by variable type

• Sign handling: Carefully track positive and negative terms

Comprehensive Summary: Simplifying Expressions
Core Concepts & Definitions:

Algebraic Expression: A mathematical phrase that can contain numbers, variables, and operation symbols.

Like Terms: Terms that have the same variable parts raised to the same powers.

Unlike Terms: Terms that have different variables or different powers.

Coefficient: The numerical factor in a term.

Constant: A term with no variable part.

Distributive Property: a(b + c) = ab + ac

Factoring: Writing an expression as a product of factors.

Greatest Common Factor (GCF): The largest factor that divides all terms.

Core Rules & Properties:

Essential Properties:

  • Combining Like Terms: Add or subtract coefficients while keeping variables unchanged
  • Distributive Property: Multiply each term inside parentheses by the factor outside
  • Factoring: Identify and extract the GCF from all terms
  • Order of Operations: Perform distribution before combining like terms

Key Formulas:

  • Distributive: a(b + c) = ab + ac
  • Factoring: ab + ac = a(b + c)
  • Like terms: ax + bx = (a + b)x
Step-by-Step Simplification Process:
  1. Identify like terms: Look for terms with identical variable parts
  2. Distribute if necessary: Remove parentheses by multiplying
  3. Group like terms: Arrange similar terms together
  4. Combine coefficients: Add or subtract the numerical parts
  5. Factor if possible: Extract common factors
  6. Verify: Check that the simplified expression is equivalent to the original
Examples & Applications:

Simple Like Terms Example:

  • 3x + 5x = (3 + 5)x = 8x

Distribution Example:

  • 2(3x + 4) = 2 × 3x + 2 × 4 = 6x + 8

Factoring Example:

  • 6x + 9 = 3(2x) + 3(3) = 3(2x + 3)

Complex Example:

  • 3x + 2(4x - 1) - x = 3x + 8x - 2 - x = 10x - 2
Tips, Tricks & Common Pitfalls:

Tips & Tricks:

  • Always check that variables match exactly before combining terms
  • When distributing, multiply by every term inside the parentheses
  • Factor out the GCF for complete simplification
  • Pay special attention to negative signs when distributing
  • Verify your answer by substituting values for variables

Common Pitfalls:

  • Combining unlike terms (e.g., 3x + 2y ≠ 5xy)
  • Forgetting to distribute to all terms in parentheses
  • Making sign errors when distributing negative numbers
  • Not factoring out the complete GCF
  • Incorrectly applying the order of operations
Key Notes for Memorization:
  • Only like terms can be combined
  • Distribution: Multiply outside factor by each term inside
  • Factoring: Reverse of distribution
  • Always verify by substituting values
  • Handle negative signs carefully
  • Factor out the greatest common factor
  • Simplified form is the most compact equivalent expression
Additional Expression Simplification Practice
a(b+c) = ab + ac
Distributive Property
Key definitions:

Algebraic expression: A combination of variables, numbers, and operations

Like terms: Terms with identical variable parts

Simplification: Reducing an expression to its most basic form

Simplification methodology:
  1. Identify: Find like terms and parentheses
  2. Distribute: Remove parentheses using distributive property
  3. Combine: Add or subtract coefficients of like terms
  4. Factor: Extract common factors if possible
  5. Verify: Check equivalence by substitution
Tip 1: Always arrange like terms together for easier combination.
Tip 2: Use parentheses to keep track of signs during distribution.
Tip 3: Factor out the complete GCF for full simplification.
Tip 4: Verify your answer by substituting values for variables.
Common errors: Combining unlike terms, distribution mistakes, sign errors.
Success strategies: Systematic approach, verification, careful sign handling.
Essential concepts:

• Like terms: Same variable parts with same exponents

• Distribution: Multiply outside factor by each term inside

• Factoring: Reverse of distribution

• Simplification: Express in most compact form

Questions & Answers

Question: How do I know if terms are "like terms" that can be combined?

Answer: Like terms must have identical variable parts with the same exponents:

  • Like terms: 3x and 5x (same variable x with exponent 1), 2x² and 4x² (same variable x with exponent 2)
  • Unlike terms: 3x and 5y (different variables), 2x and 3x² (same variable but different exponents)
  • Constants: Numbers without variables (like 5 and 7) are also like terms

Only like terms can be combined by adding or subtracting their coefficients.

Question: When distributing a negative number, how do I handle the signs?

Answer: When distributing a negative number, multiply each term inside the parentheses by the negative number:

  • Example: -3(2x - 4) = -3 × 2x + (-3) × (-4) = -6x + 12
  • Rule: A negative times a positive equals a negative
  • Rule: A negative times a negative equals a positive

Every term inside the parentheses gets multiplied by the negative number, changing each sign accordingly.

Question: What's the difference between simplifying and solving an equation?

Answer: The key differences are:

  • Simplifying: Rewriting an expression in a more compact form without solving for a variable (e.g., 3x + 2x becomes 5x)
  • Solving: Finding the value(s) of the variable that make an equation true (e.g., 3x + 2 = 11, so x = 3)

Simplification is often a step in solving equations, but they are distinct processes with different goals.