Combining Like Terms & Distributive Property

Combining like terms and the distributive property are fundamental concepts in algebra, serving as the building blocks for solving equations, simplifying expressions, and tackling more complex mathematical problems. Mastering these skills is crucial for students and anyone who wants to gain proficiency in mathematics.

Understanding Like Terms

Like terms are terms that contain the same variable(s) raised to the same power. Only the coefficients (the numbers in front of the variables) can be different.

Identifying Like Terms

To identify like terms, look for terms that have:

• The same variables: For example, 3x and -5x are like terms because they both have the variable x.
• The same exponents: For example, 2x^2 and 7x^2 are like terms because they both have the variable x raised to the power of 2.

Examples of Like Terms:

• 4y and -9y
• 12ab and -2ab
• 5x^3 and -x^3
• 8 and -3 (Constants are like terms)

Examples of Unlike Terms:

• 3x and 4y (Different variables)
• 2x^2 and 5x (Different exponents)
• 7ab and 7a (Missing the variable b in the second term)

Combining Like Terms

Combining like terms means simplifying an expression by adding or subtracting the coefficients of like terms. Here's the process:

1. Identify like terms in the expression.
2. Combine the coefficients of the like terms by adding or subtracting them.
3. Write the result with the common variable and exponent.

Examples of Combining Like Terms:

• Example 1: Simplify 3x + 5x - 2x

  • All terms are like terms because they all have the variable x raised to the power of 1.
  • Combine the coefficients: 3 + 5 - 2 = 6
  • The simplified expression is 6x.

• Example 2: Simplify 4y^2 - 2y^2 + y^2 + 6y - y

  • Identify like terms: 4y^2, -2y^2, and y^2 are like terms. Also, 6y and -y are like terms.
  • Combine the coefficients: For the y^2 terms: 4 - 2 + 1 = 3. For the y terms: 6 - 1 = 5.
  • The simplified expression is 3y^2 + 5y.

• Example 3: Simplify 7a + 3b - 2a + 5b - 4

  • Identify like terms: 7a and -2a are like terms. Also, 3b and 5b are like terms. -4 is a constant term and has no like terms in this expression.
  • Combine the coefficients: For the a terms: 7 - 2 = 5. For the b terms: 3 + 5 = 8.
  • The simplified expression is 5a + 8b - 4.

Tips for Combining Like Terms

• Pay attention to the signs: Make sure to include the signs (+ or -) when combining the coefficients.
• Only combine like terms: You cannot combine terms that are not like terms. For example, 3x and 4y cannot be combined.
• Write the simplified expression in a standard form: Typically, terms are arranged in descending order of their exponents. For example, 3x^2 + 5x - 2 is in standard form.
• Use highlighters or different colored pens: When dealing with lengthy expressions, using different colors to highlight like terms can help prevent mistakes.
• Double-check your work: Before moving on, ensure you've correctly identified and combined all like terms.

Understanding the Distributive Property

The distributive property is a fundamental algebraic property that allows you to multiply a single term by two or more terms inside a set of parentheses. It's a powerful tool for simplifying expressions and solving equations.

The Distributive Property Formula

The distributive property can be expressed as:

a(b + c) = ab + ac

where a, b, and c represent any real numbers or algebraic terms.

In simpler terms, to use the distributive property, you multiply the term outside the parentheses (a) by each term inside the parentheses (b and c), and then add the results.

Applying the Distributive Property

Here's how to apply the distributive property:

1. Identify the term outside the parentheses and the terms inside the parentheses.
2. Multiply the term outside the parentheses by each term inside the parentheses.
3. Simplify the resulting expression by combining like terms, if possible.

Examples of Applying the Distributive Property:

• Example 1: Simplify 2(x + 3)

  • The term outside the parentheses is 2, and the terms inside are x and 3.
  • Multiply 2 by x: 2 * x = 2x
  • Multiply 2 by 3: 2 * 3 = 6
  • The simplified expression is 2x + 6.

• Example 2: Simplify -3(2y - 5)

  • The term outside the parentheses is -3, and the terms inside are 2y and -5.
  • Multiply -3 by 2y: -3 * 2y = -6y
  • Multiply -3 by -5: -3 * -5 = 15
  • The simplified expression is -6y + 15.

• Example 3: Simplify x(x - 4)

  • The term outside the parentheses is x, and the terms inside are x and -4.
  • Multiply x by x: x * x = x^2
  • Multiply x by -4: x * -4 = -4x
  • The simplified expression is x^2 - 4x.

• Example 4: Simplify 5(a + 2b - c)

  • The term outside the parentheses is 5, and the terms inside are a, 2b, and -c.
  • Multiply 5 by a: 5 * a = 5a
  • Multiply 5 by 2b: 5 * 2b = 10b
  • Multiply 5 by -c: 5 * -c = -5c
  • The simplified expression is 5a + 10b - 5c.

Distributing with Negative Signs

Be particularly careful when distributing with negative signs. Remember that multiplying a negative number by a positive number results in a negative number, and multiplying a negative number by a negative number results in a positive number.

• Example: Simplify -(x - 7)

  • Think of the negative sign as -1 being multiplied by the parentheses: -1(x - 7)
  • Multiply -1 by x: -1 * x = -x
  • Multiply -1 by -7: -1 * -7 = 7
  • The simplified expression is -x + 7.

Tips for Using the Distributive Property

• Pay close attention to signs: Double-check the signs when multiplying, especially with negative numbers.
• Distribute to every term: Make sure to multiply the term outside the parentheses by every term inside the parentheses.
• Combine like terms after distributing: After applying the distributive property, look for like terms and combine them to simplify the expression further.
• Practice regularly: The more you practice using the distributive property, the more comfortable and confident you'll become.

Combining Like Terms and the Distributive Property Together

Often, you'll encounter expressions that require you to use both the distributive property and combining like terms. This involves a two-step process:

1. Apply the distributive property to remove the parentheses.
2. Combine like terms to simplify the resulting expression.

Examples of Combining Like Terms and the Distributive Property:

• Example 1: Simplify 3(x + 2) + 4x

  • Apply the distributive property: 3 * x = 3x and 3 * 2 = 6. This gives us 3x + 6 + 4x.
  • Combine like terms: 3x and 4x are like terms. 3x + 4x = 7x.
  • The simplified expression is 7x + 6.

• Example 2: Simplify 2(y - 1) - 5(y + 3)

  • Apply the distributive property to the first set of parentheses: 2 * y = 2y and 2 * -1 = -2. This gives us 2y - 2.
  • Apply the distributive property to the second set of parentheses: -5 * y = -5y and -5 * 3 = -15. This gives us -5y - 15.
  • Now we have: 2y - 2 - 5y - 15.
  • Combine like terms: 2y and -5y are like terms. 2y - 5y = -3y. -2 and -15 are like terms. -2 - 15 = -17.
  • The simplified expression is -3y - 17.

• Example 3: Simplify 4(a + b) - 2(a - b) + 3a

  • Apply the distributive property to the first set of parentheses: 4 * a = 4a and 4 * b = 4b. This gives us 4a + 4b.
  • Apply the distributive property to the second set of parentheses: -2 * a = -2a and -2 * -b = 2b. This gives us -2a + 2b.
  • Now we have: 4a + 4b - 2a + 2b + 3a.
  • Combine like terms: 4a, -2a, and 3a are like terms. 4a - 2a + 3a = 5a. 4b and 2b are like terms. 4b + 2b = 6b.
  • The simplified expression is 5a + 6b.

Common Mistakes to Avoid

• Forgetting to distribute to all terms: Ensure you multiply the term outside the parentheses by every term inside.
• Incorrectly applying the signs: Pay careful attention to negative signs when distributing.
• Combining unlike terms: Only combine terms that have the same variable(s) raised to the same power.
• Not simplifying completely: After distributing and combining like terms, double-check to see if there are any further simplifications possible.

Advanced Applications

Combining like terms and the distributive property are not just isolated skills; they are essential tools for solving more advanced algebraic problems, including:

• Solving equations: Simplifying both sides of an equation using these techniques is often the first step in isolating the variable and finding the solution.
• Factoring: The distributive property is the basis for factoring, which is the process of breaking down an expression into its factors.
• Working with polynomials: Polynomials are expressions with multiple terms, and combining like terms and the distributive property are used extensively to simplify and manipulate them.
• Calculus: These skills are foundational for calculus, particularly when dealing with derivatives and integrals.

Practice Problems

Here are some practice problems to help you solidify your understanding of combining like terms and the distributive property:

1. Simplify: 5x - 2x + 7x - x
2. Simplify: 3y^2 + 4y - y^2 - 2y + 5
3. Simplify: 8a - 3b + 2a + 5b - 4a + b
4. Simplify: 2(x + 5) - 3x
5. Simplify: -4(2y - 1) + 6y - 3
6. Simplify: x(x - 3) + 2x^2 - 5x
7. Simplify: 5(a + 2b) - 3(2a - b) + a - 4b
8. Simplify: -(3x - 4) + 2(x + 1) - 5

Answer Key:

1. 9x
2. 2y^2 + 2y + 5
3. 6a + 3b
4. -x + 10
5. -2y + 1
6. 3x^2 - 8x
7. 4b
8. -x + 1

Conclusion

Mastering combining like terms and the distributive property is essential for success in algebra and beyond. By understanding the concepts, practicing regularly, and avoiding common mistakes, you can build a strong foundation in mathematics. These skills will empower you to tackle more complex problems with confidence and ease. Remember to break down complex problems into smaller steps, double-check your work, and never be afraid to ask for help when needed. With dedication and practice, you can master these fundamental algebraic concepts and unlock your mathematical potential.