Solved Exercises on Combining Like Terms in Grade 9

Master combining like terms: identifying terms, coefficients, and simplifying expressions through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Basic Like Terms
Exercise 1
Simplify the expression: 3x + 5x - 2x + 7
Definition:

Like terms: Terms with the same variable part (same variable and exponent)

Constant term: A term with no variable (just a number)

Coefficient: The number multiplied by the variable

Combining like terms method:
  1. Identify terms with the same variable part
  2. Group like terms together
  3. Add or subtract the coefficients
  4. Keep the variable part unchanged
  5. Combine constant terms separately
Original
3x + 5x - 2x + 7
Group like terms
(3x + 5x - 2x) + 7
Simplified
6x + 7
Step 1: Identify like terms

Terms with variable x: 3x, 5x, -2x

Constant term: 7

Step 2: Group like terms

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

Step 3: Add the coefficients

3 + 5 - 2 = 6

Step 4: Write the simplified expression

6x + 7

6x + 7
Final answer:

The simplified expression is 6x + 7

Applied rules:

• Like terms: Same variable parts can be combined

• Coefficient addition: Add coefficients while keeping variables

• Constant terms: Remain unchanged

2 Multiple Variables
Exercise 2
Simplify the expression: 4x + 3y - 2x + 5y - 6
Definition:

Multiple variables: Expressions with more than one type of variable

Like terms rule: Only terms with identical variable parts can be combined

Unlike terms: Cannot be combined (different variables)

Original
4x + 3y - 2x + 5y - 6
Group like terms
(4x - 2x) + (3y + 5y) - 6
Simplified
2x + 8y - 6
Step 1: Identify terms by variable

Terms with x: 4x, -2x

Terms with y: 3y, 5y

Constant term: -6

Step 2: Group like terms

(4x - 2x) + (3y + 5y) - 6

Step 3: Combine coefficients for each variable

For x: 4 - 2 = 2

For y: 3 + 5 = 8

Step 4: Write the simplified expression

2x + 8y - 6

2x + 8y - 6
Final answer:

The simplified expression is 2x + 8y - 6

Applied rules:

• Variable grouping: Only combine terms with identical variables

• Independent variables: x and y terms cannot be combined

• Constant terms: Combine separately

3 Unlike Terms
Exercise 3
Explain why 3x² and 5x cannot be combined. Simplify 3x² + 5x - x² + 2x.
Definition:

Unlike terms: Terms with different variable parts (different variables or exponents)

Exponent matters: x² and x have different variable parts

Cannot combine: Unlike terms remain separate in simplified expressions

Original
3x² + 5x - x² + 2x
Group like terms
(3x² - x²) + (5x + 2x)
Simplified
2x² + 7x
Step 1: Identify why 3x² and 5x are unlike terms

3x² has variable part x²

5x has variable part x

Since x² ≠ x, these are unlike terms and cannot be combined

Step 2: For the expression, identify like terms

Terms with x²: 3x², -x²

Terms with x: 5x, 2x

Step 3: Group like terms

(3x² - x²) + (5x + 2x)

Step 4: Combine coefficients

For x²: 3 - 1 = 2

For x: 5 + 2 = 7

Step 5: Write the simplified expression

2x² + 7x

2x² + 7x
Final answer:

3x² and 5x cannot be combined because they have different variable parts. The simplified expression is 2x² + 7x

Applied rules:

• Variable parts: Both variable and exponent must match

• Exponent matching: x² and x are unlike terms

• Grouping: Only combine terms with identical variable parts

Like Terms Fundamentals
\(ax^n + bx^n = (a+b)x^n, \text{ where } a, b \text{ are coefficients and } n \text{ is exponent}\)
Like Terms Formula
Like Terms
3x + 5x = 8x
Same variable part
Unlike Terms
3x + 5y ≠ 8xy
Different variables
Different Exponents
x² + x ≠ 2x³
Cannot combine
Key definitions:

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

Coefficient: The numerical factor in a term

Variable part: The variable and its exponent in a term

Like terms: Terms with identical variable parts

Combining like terms methodology:
  1. Identify terms: Recognize each term in the expression
  2. Classify by variable: Group terms with same variable parts
  3. Add coefficients: Sum the numerical factors
  4. Preserve variables: Keep variable parts unchanged
  5. Write result: Combine all simplified terms
Tip 1: Like terms must have the same variable AND the same exponent.
Tip 2: Always check the exponent - x² and x are unlike terms.
Tip 3: Constants (numbers without variables) can be combined with other constants.
Tip 4: If terms are unlike, they remain separate in the simplified expression.
Common errors: Combining unlike terms, ignoring exponents, sign errors, calculation mistakes.
Exam preparation: Practice identifying like terms, memorize the rules, work with multiple variables.
Essential rules:

• Like terms: Only terms with identical variable parts can be combined

• Coefficient addition: Add coefficients while keeping variable part

• Unlike terms: Remain separate in simplified expressions

• Exponent matching: x² and x are unlike terms

Solution: Exercises 4 to 5
4 Distribution and Combining
Exercise 4
Simplify: 2(3x + 4) + 5x - 3(2x - 1)
Definition:

Distribution: Multiplying a factor across terms in parentheses

Order of operations: Distribute first, then combine like terms

Sign awareness: Pay attention to signs when distributing

Original
2(3x + 4) + 5x - 3(2x - 1)
Distribute
6x + 8 + 5x - 6x + 3
Simplify
5x + 11
Step 1: Distribute the first set of parentheses

2(3x + 4) = 2 × 3x + 2 × 4 = 6x + 8

Step 2: Distribute the second set of parentheses

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

Step 3: Rewrite the expression after distribution

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

Step 4: Group like terms

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

Step 5: Combine like terms

For x terms: 6 + 5 - 6 = 5

For constants: 8 + 3 = 11

Step 6: Write the simplified expression

5x + 11

5x + 11
Final answer:

The simplified expression is 5x + 11

Applied rules:

• Distribution: Multiply factor by each term in parentheses

• Sign handling: Negative factor changes signs of terms

• Combining like terms: After distribution

5 Complex Expression
Exercise 5
Simplify: 4x² - 3xy + 2y² + x² + 5xy - y² - 2x² + xy
Definition:

Multiple variable terms: Terms with different combinations of variables

Classification: Group by variable parts (x², xy, y²)

Complex expressions: Require careful organization and attention to detail

Original
4x² - 3xy + 2y² + x² + 5xy - y² - 2x² + xy
Group by type
(4x² + x² - 2x²) + (-3xy + 5xy + xy) + (2y² - y²)
Simplify
3x² + 3xy + y²
Step 1: Identify and classify terms by variable part

Terms with x²: 4x², x², -2x²

Terms with xy: -3xy, 5xy, xy

Terms with y²: 2y², -y²

Step 2: Group like terms together

(4x² + x² - 2x²) + (-3xy + 5xy + xy) + (2y² - y²)

Step 3: Combine coefficients for each group

For x² terms: 4 + 1 - 2 = 3

For xy terms: -3 + 5 + 1 = 3

For y² terms: 2 - 1 = 1

Step 4: Write the simplified expression

3x² + 3xy + y²

3x² + 3xy + y²
Final answer:

The simplified expression is 3x² + 3xy + y²

Applied rules:

• Multi-variable classification: Group by exact variable parts

• Organized approach: Identify all term types first

• Systematic combination: Work through each group separately

Detailed Summary: Combining Like Terms
\(ax^n + bx^n = (a+b)x^n, \text{ where } a, b \text{ are coefficients and } n \text{ is exponent}\)
Like Terms Combination Formula
Comprehensive definitions:

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

Coefficient: The numerical factor in a term (the number multiplied by the variable)

Variable part: The variable and its exponent in a term (e.g., x², xy, y³)

Like terms: Terms that have identical variable parts (same variables and same exponents)

Unlike terms: Terms that have different variable parts and cannot be combined

Complete combining like terms methodology:
  1. Term identification: Recognize each term in the expression
  2. Variable classification: Group terms by identical variable parts
  3. Coefficient calculation: Add or subtract the coefficients of like terms
  4. Variable preservation: Keep the variable part unchanged
  5. Result compilation: Combine all simplified terms
Tip 1: Remember that terms like x, x², and x³ are all unlike terms and cannot be combined.
Tip 2: When distributing, pay close attention to signs - negative factors change signs of terms.
Tip 3: Organize complex expressions by drawing lines or using different colors for different term types.
Tip 4: Always check that your final expression has no like terms that can be further combined.
Common applications: Simplifying expressions, solving equations, factoring polynomials, calculus preparation.
Key skills: Pattern recognition, algebraic manipulation, attention to detail, systematic approach.
Essential rules and properties:

• Like terms rule: Only terms with identical variable parts can be combined

• Coefficient addition: Add coefficients while preserving variable part

• Exponent matching: x² and x are unlike terms, cannot combine

• Sign preservation: Negative coefficients are added as negatives

• Constant terms: Combine separately from variable terms

• Order independence: Terms can be rearranged for easier grouping

Visualization: Like Terms Concepts
Term Classification
Visualizing like terms and their classification:
Same variable parts, different variable parts
How coefficients change during combination

Analysis: The chart shows how like terms can be combined while unlike terms remain separate.

  • Like terms have the same variable parts and can be combined
  • Unlike terms remain separate in simplified expressions
  • Coefficients are added while variables remain unchanged

Questions & Answers

Question: Can I combine x and x²? They both have the variable x, shouldn't they be like terms?

Answer: No, x and x² are NOT like terms. For terms to be like terms, they must have identical variable parts, which means the same variables raised to the same powers.

Here's the difference:

  • x: Has variable part x¹ (or just x)
  • x²: Has variable part x²

Since x¹ ≠ x², these are unlike terms and cannot be combined. Think of it like this: if x represents apples, then x² represents apple squares (which doesn't make sense in reality), so you can't combine them.

Like terms: 3x and 5x, 2x² and -4x², 7xy and -2xy

Unlike terms: x and x², x and y, 3x² and 2x

Question: What happens when I have terms like 3xy and 5yx? Are they like terms?

Answer: Yes, 3xy and 5yx are like terms! This is because multiplication is commutative, meaning xy = yx.

The variable parts are identical:

  • 3xy has variable part xy
  • 5yx has variable part yx, which equals xy

So: 3xy + 5yx = 3xy + 5xy = 8xy

Remember that the order of variables doesn't matter due to the commutative property of multiplication. However, be careful with exponents: x²y and xy² are unlike terms because the exponents are on different variables.

Question: How do I handle negative coefficients when combining like terms?

Answer: Treat negative coefficients as negative numbers in the addition process:

For example: 5x - 3x + 2x

This is equivalent to: 5x + (-3x) + 2x

Combine coefficients: 5 + (-3) + 2 = 4

Result: 4x

Another example: 7y - 9y

Coefficients: 7 + (-9) = -2

Result: -2y

Always pay attention to the sign that comes before each term when combining like terms.

Question: What if I have a term with no visible coefficient, like x?

Answer: When a variable appears with no visible coefficient, the coefficient is understood to be 1.

For example:

  • x = 1x
  • -x = -1x
  • xy = 1xy

So when combining x + 3x, you're actually combining 1x + 3x = 4x

And when combining 5y - y, you're combining 5y - 1y = 4y

This rule applies to any variable that appears without a visible coefficient - the coefficient is 1.

Question: How do I know when I'm done simplifying an expression?

Answer: An expression is fully simplified when:

  • All like terms have been combined
  • There are no parentheses that need to be removed
  • No further arithmetic can be performed

To check if you're done:

  1. Scan the expression for any terms that have identical variable parts
  2. If you find any, you need to combine them
  3. Make sure all distributions have been completed
  4. Verify that no arithmetic errors were made

For example, 3x + 2y + 5 is fully simplified because x-terms, y-terms, and constants are all different and cannot be combined further.