Solved Exercises on Combining Like Terms in Grade 7

Master combining like terms: identifying like terms, adding and subtracting coefficients, simplifying expressions, and solving equations through these 10 detailed exercises with visual learning tools.

Solution: Exercises 1 to 3
1 Basic Like Terms
Exercise 1
Simplify: 3x + 5x - 2x
Definition:

Like terms: Terms that have the same variable part with the same exponent.

Method:
  1. Identify terms with the same variable part
  2. Add or subtract the coefficients (numbers in front)
  3. Keep the variable part unchanged
Step 1: Identify like terms

All terms have the same variable: x

Step 2: Add/subtract coefficients

3 + 5 - 2 = 6

Step 3: Write the simplified expression

6x

6x
Final answer:

3x + 5x - 2x simplifies to 6x.

Applied rules:

• Like terms: Only combine terms with identical variable parts

• Coefficient arithmetic: Add/subtract numbers in front of variables

• Variable preservation: Keep variable part unchanged

2 Mixed Like Terms
Exercise 2
Simplify: 4x + 3y - 2x + 5y + 7
Definition:

Unlike terms: Terms with different variables or different exponents cannot be combined.

Step 1: Group like terms together

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

Step 2: Combine like terms

(4 - 2)x + (3 + 5)y + 7 = 2x + 8y + 7

Step 3: State the final answer

2x + 8y + 7

Grouping by variable:
4x
-2x
→
2x
3y
5y
→
8y
7
→
7
2x + 8y + 7
Final answer:

4x + 3y - 2x + 5y + 7 simplifies to 2x + 8y + 7.

Applied rules:

• Grouping: Organize terms by variable type

• Separate operations: Combine only terms with identical variables

• Constants: Keep numbers separate from variables

3 Complex Like Terms
Exercise 3
Simplify: 5x² + 3x - 2x² + 4x - 7 + x²
Definition:

Like terms with exponents: Terms with the same variable and same exponent.

Step 1: Group like terms by exponent

(5x² - 2x² + x²) + (3x + 4x) + (-7)

Step 2: Combine terms with x²

(5 - 2 + 1)x² = 4x²

Step 3: Combine terms with x¹

(3 + 4)x = 7x

Step 4: Keep the constant

Constant: -7

Step 5: Write the simplified expression

4x² + 7x - 7

Grouping by exponent:
5x²
-2x²
x²
→
4x²
3x
4x
→
7x
-7
→
-7
4x² + 7x - 7
Final answer:

5x² + 3x - 2x² + 4x - 7 + x² simplifies to 4x² + 7x - 7.

Applied rules:

• Exponent matching: Only combine terms with same variable AND same exponent

• Grouping by degree: Organize by highest to lowest exponent

• Constant terms: Keep numbers separate from variables

Solution: Exercises 4 to 6
4 Subtraction with Like Terms
Exercise 4
Simplify: 8x - 3y + 2x - 5y - 4x + y
Definition:

Subtraction with like terms: Treat subtraction as adding a negative coefficient.

Step 1: Group like terms

(8x + 2x - 4x) + (-3y - 5y + y)

Step 2: Combine x terms

(8 + 2 - 4)x = 6x

Step 3: Combine y terms

(-3 - 5 + 1)y = -7y

Step 4: Write the simplified expression

6x - 7y

Grouping with signs:
8x
2x
-4x
→
6x
-3y
-5y
y
→
-7y
6x - 7y
Final answer:

8x - 3y + 2x - 5y - 4x + y simplifies to 6x - 7y.

Applied rules:

• Sign handling: Include the sign with the coefficient

• Arithmetic with negatives: Be careful with addition and subtraction

• Grouping: Combine terms with identical variables separately

5 Multiple Exponents
Exercise 5
Simplify: 3x³ + 2x² - x + 5x³ - x² + 4x - 2
Definition:

Terms with different exponents: Only combine terms with the same variable and same exponent.

Step 1: Group like terms by exponent

(3x³ + 5x³) + (2x² - x²) + (-x + 4x) + (-2)

Step 2: Combine x³ terms

(3 + 5)x³ = 8x³

Step 3: Combine x² terms

(2 - 1)x² = 1x² = x²

Step 4: Combine x terms

(-1 + 4)x = 3x

Step 5: Keep the constant

Constant: -2

Step 6: Write the simplified expression

8x³ + x² + 3x - 2

Grouping by degree:
3x³
5x³
→
8x³
2x²
-x²
→
x²
-x
4x
→
3x
-2
→
-2
8x³ + x² + 3x - 2
Final answer:

3x³ + 2x² - x + 5x³ - x² + 4x - 2 simplifies to 8x³ + x² + 3x - 2.

Applied rules:

• Degree matching: Only combine terms with same variable and same exponent

• Descending order: Arrange terms from highest to lowest degree

• Sign tracking: Keep track of positive and negative coefficients

6 Reversing Operations
Exercise 6
If 4x + 2y - 3x + 5y simplifies to ax + by, what are the values of a and b?
Definition:

Variable coefficient identification: Finding the coefficient of each variable after simplification.

Step 1: Group like terms

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

Step 2: Combine x terms

(4 - 3)x = 1x = x

Step 3: Combine y terms

(2 + 5)y = 7y

Step 4: Identify coefficients

1x + 7y = ax + by, so a = 1 and b = 7

Coefficient identification:
4x
-3x
→
1x
2y
5y
→
7y
a = 1, b = 7
Final answer:

The values are a = 1 and b = 7.

Applied rules:

• Coefficient extraction: The number in front of each variable is the coefficient

• Variable matching: Match simplified terms to target form

• Equation comparison: Compare like terms on both sides

Like Terms Visual Guide
ax + bx = (a+b)x
Like Terms Formula
Variable x
Coefficient changes
Variable y
Different from x terms
Exponents
Must match exactly
Constants
Numbers without variables
Like Terms Identification Process:
Step 1: Look for terms with identical variables
Step 2: Check that exponents match exactly
Step 3: Group terms with same variable parts
Step 4: Add coefficients of like terms
Step 5: Keep variable part unchanged
Tip 1: Like terms must have identical variable parts including exponents.
Tip 2: Always include the sign with the coefficient when adding.
Tip 3: Constants (numbers without variables) are like terms with each other.
Common errors: Combining unlike terms, forgetting to include signs, mismatching exponents.
Success strategies: Group by variable first, verify variable parts match, double-check arithmetic.
Essential concepts:

• Like terms: Same variable parts with same exponents

• Coefficient: Number in front of variable

• Variable part: Letter(s) with exponents

• Addition: Add coefficients, keep variable part

Solution: Exercises 7 to 10
7 Multi-Variable Expression
Exercise 7
Simplify: 2x + 3y - z + 4x - y + 2z - x + 5y
Definition:

Multi-variable expression: An expression with multiple different variables.

Step 1: Group like terms by variable

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

Step 2: Combine x terms

(2 + 4 - 1)x = 5x

Step 3: Combine y terms

(3 - 1 + 5)y = 7y

Step 4: Combine z terms

(-1 + 2)z = 1z = z

Step 5: Write the simplified expression

5x + 7y + z

Grouping by variable:
2x
4x
-x
→
5x
3y
-y
5y
→
7y
-z
2z
→
z
5x + 7y + z
Final answer:

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

Applied rules:

• Multi-variable grouping: Group by each unique variable

• Separate operations: Combine each variable type independently

• Sign tracking: Include signs with coefficients

8 Mixed Exponents and Signs
Exercise 8
Simplify: -3x² + 5x - 2 + 4x² - x + 7 - 2x² + 3x
Definition:

Expression with mixed signs: An expression containing both positive and negative terms.

Step 1: Group like terms by exponent

(-3x² + 4x² - 2x²) + (5x - x + 3x) + (-2 + 7)

Step 2: Combine x² terms

(-3 + 4 - 2)x² = -1x² = -x²

Step 3: Combine x terms

(5 - 1 + 3)x = 7x

Step 4: Combine constants

-2 + 7 = 5

Step 5: Write the simplified expression

-x² + 7x + 5

Grouping with signs:
-3x²
4x²
-2x²
→
-x²
5x
-x
3x
→
7x
-2
7
→
5
-x² + 7x + 5
Final answer:

-3x² + 5x - 2 + 4x² - x + 7 - 2x² + 3x simplifies to -x² + 7x + 5.

Applied rules:

• Sign handling: Include negative signs with coefficients

• Arithmetic with negatives: Carefully add positive and negative numbers

• Exponent matching: Only combine terms with same variable and exponent

9 Complex Expression
Exercise 9
Simplify: 6a²b - 3ab² + 2a²b - ab² + 4ab - 5a²b + 2ab²
Definition:

Multi-variable terms: Terms with more than one variable, like a²b or ab².

Step 1: Group like terms by variable parts

(6a²b + 2a²b - 5a²b) + (-3ab² - ab² + 2ab²) + (4ab)

Step 2: Combine a²b terms

(6 + 2 - 5)a²b = 3a²b

Step 3: Combine ab² terms

(-3 - 1 + 2)ab² = -2ab²

Step 4: Keep the ab term

4ab remains as is

Step 5: Write the simplified expression

3a²b - 2ab² + 4ab

Grouping by variable parts:
6a²b
2a²b
-5a²b
→
3a²b
-3ab²
-ab²
2ab²
→
-2ab²
4ab
→
4ab
3a²b - 2ab² + 4ab
Final answer:

6a²b - 3ab² + 2a²b - ab² + 4ab - 5a²b + 2ab² simplifies to 3a²b - 2ab² + 4ab.

Applied rules:

• Variable part matching: Terms like a²b and ab² are different

• Multi-variable grouping: Group by identical variable combinations

• Complex expressions: Focus on variable parts, not just individual letters

10 Advanced Problem
Exercise 10
Simplify: 4x²y - 2xy² + 3xy - x²y + 5xy² - 2xy + 6x²y - xy² + xy
Definition:

Advanced multi-variable expression: An expression with several terms of different variable combinations.

Step 1: Group like terms by variable parts

(4x²y - x²y + 6x²y) + (-2xy² + 5xy² - xy²) + (3xy - 2xy + xy)

Step 2: Combine x²y terms

(4 - 1 + 6)x²y = 9x²y

Step 3: Combine xy² terms

(-2 + 5 - 1)xy² = 2xy²

Step 4: Combine xy terms

(3 - 2 + 1)xy = 2xy

Step 5: Write the simplified expression

9x²y + 2xy² + 2xy

Grouping by variable parts:
4x²y
-x²y
6x²y
→
9x²y
-2xy²
5xy²
-xy²
→
2xy²
3xy
-2xy
xy
→
2xy
9x²y + 2xy² + 2xy
Final answer:

4x²y - 2xy² + 3xy - x²y + 5xy² - 2xy + 6x²y - xy² + xy simplifies to 9x²y + 2xy² + 2xy.

Applied rules:

• Systematic grouping: Organize by identical variable parts

• Sign tracking: Carefully handle positive and negative coefficients

• Verification: Check that all original terms are accounted for

Comprehensive Summary: Combining Like Terms
Core Concepts & Definitions:

Like Terms: Terms that have identical variable parts with the same exponents. For example, 3x and 5x are like terms, but 3x and 3x² are not.

Unlike Terms: Terms with different variable parts or different exponents that cannot be combined. For example, 3x and 4y are unlike terms.

Coefficient: The numerical factor in a term. In 7x, the coefficient is 7.

Variable Part: The letter(s) in a term with their exponents. In 7x², the variable part is x².

Constant: A term with no variable part. For example, 5 is a constant.

Combining Like Terms: The process of adding or subtracting coefficients of like terms while keeping the variable part unchanged.

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

Simplification: The process of writing an expression in its most compact form.

Core Rules & Properties:

Essential Properties:

  • Variable Matching: Only terms with identical variable parts can be combined
  • Exponent Matching: The exponents must be exactly the same
  • Coefficient Arithmetic: Add or subtract the numerical coefficients
  • Variable Preservation: The variable part remains unchanged

Key Rules:

  • ax + bx = (a + b)x (combining like terms)
  • ax - bx = (a - b)x (subtracting like terms)
  • Terms with different variables cannot be combined
  • Constants can be combined with other constants
Step-by-Step Process:
  1. Identify like terms: Look for terms with identical variable parts
  2. Group like terms: Rearrange the expression to group similar terms together
  3. Check exponents: Ensure exponents match exactly
  4. Add coefficients: Add or subtract the numerical parts
  5. Preserve variables: Keep the variable part unchanged
  6. Write simplified form: Express the result in standard order
Examples & Applications:

Simple Example:

  • 3x + 5x = (3 + 5)x = 8x

Two Variables:

  • 2x + 3y - x + 4y = (2x - x) + (3y + 4y) = x + 7y

With Exponents:

  • 4x² + 3x - 2x² + x = (4x² - 2x²) + (3x + x) = 2x² + 4x

Complex Example:

  • 5a²b - 2ab² + 3a²b - ab² = (5a²b + 3a²b) + (-2ab² - ab²) = 8a²b - 3ab²
Tips, Tricks & Common Pitfalls:

Tips & Tricks:

  • Always include the sign with the coefficient when adding
  • Group terms by variable type to avoid missing like terms
  • Check that variables AND exponents match exactly
  • Rewrite expressions with like terms adjacent to make combining easier
  • Remember that x is the same as 1x and -x is the same as -1x

Common Pitfalls:

  • Combining unlike terms (e.g., 3x + 4y ≠ 7xy)
  • Forgetting to include the negative sign with coefficients
  • Mistaking terms with different exponents as like terms
  • Miscalculating with negative numbers
  • Changing the variable part when combining terms
Key Notes for Memorization:
  • Like terms: Same variable parts with same exponents
  • Add coefficients, keep variables unchanged
  • Include the sign with the coefficient
  • Unlike terms cannot be combined
  • Constants combine with constants
  • Arrange in descending order by exponent
  • Always verify all original terms are accounted for
Additional Like Terms Practice
ax + bx = (a+b)x
Like Terms Formula
Key definitions:

Like terms: Terms with identical variable parts and exponents

Combining: Adding or subtracting coefficients while preserving variables

Simplification: Writing expressions in most compact form

Combining methodology:
  1. Identify: Find terms with identical variable parts
  2. Group: Organize like terms together
  3. Calculate: Add or subtract coefficients
  4. Preserve: Keep variable parts unchanged
  5. Verify: Ensure all terms are accounted for
Tip 1: Always include the sign with the coefficient when adding.
Tip 2: Check that variables AND exponents match exactly.
Tip 3: Group terms by variable type to avoid mistakes.
Tip 4: Remember x = 1x and -x = -1x.
Common errors: Combining unlike terms, sign mistakes, exponent mismatches.
Success strategies: Systematic grouping, verification, careful sign handling.
Essential concepts:

• Like terms: Same variable parts

• Coefficient: Number in front

• Variable: Letter part unchanged

• Constants: Combine with constants

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: What's the difference between 3x² and (3x)²? Can they be combined?

Answer: These are completely different expressions and cannot be combined:

  • 3x²: The coefficient 3 multiplies x² (so it's 3 × x²)
  • (3x)²: This means (3x) × (3x) = 9x²

Even though both simplify to a term with x², they have different coefficients (3 and 9) and come from different operations.

Question: Can I combine x and x² terms since they both have the variable x?

Answer: No, x and x² are not like terms and cannot be combined:

  • x: Represents x to the first power (x¹)
  • x²: Represents x to the second power
  • Key rule: Both the variable AND the exponent must be identical to be like terms

Think of it this way: x is like "one apple" and x² is like "one apple squared" - they're fundamentally different quantities.