What Are Like Terms?

Like terms are terms in an algebraic expression that have the same variable raised to the same power. They can have different coefficients, which are numerical constants, but the variables and their exponents must match exactly. For example, 3x and –2x are like terms because they both contain the variable x raised to the first power. Similarly, 4y² and 5y² are like terms. However, 3x and 3x² are not like terms because the exponents differ. Additionally, constants, such as 7 and –12, are also considered like terms since they have no variables. Understanding and identifying like terms is essential for combining them to simplify expressions and solve equations effectively.

What Is the Distributive Property?

The distributive property is a fundamental algebraic principle that allows you to simplify expressions by multiplying a single term outside parentheses with each term inside. It is often written as a(b + c) = ab + ac. This property helps break down complex expressions into simpler, more manageable parts. For example, in the expression 3(x + 4), the distributive property would be applied by multiplying 3 by both x and 4, resulting in 3x + 12. After applying the distributive property, the next step is often to combine like terms to further simplify the expression. This property is essential for solving equations and simplifying expressions efficiently in algebra. Mastering it is a key step in developing strong mathematical problem-solving skills.

Applying Both Concepts Together

Applying the distributive property and combining like terms together simplifies complex algebraic expressions. First, apply the distributive property to expand expressions, then combine like terms to achieve the simplest form.

Step-by-Step Guide to Using Both Concepts

To effectively use the distributive property and combine like terms, follow these steps:

1. Identify Like Terms: Determine which terms in the expression have the same variable and exponent.
2. Apply the Distributive Property: Multiply the outside number or variable by each term inside the parentheses.



3. Combine Like Terms: Add or subtract the coefficients of like terms to simplify the expression.
4. Simplify: Write the final simplified expression after combining like terms.

For example, simplify (6x ⸺ 2(x + 4)):

• Distribute: (6x ⸺ 2x ⸺ 8)
• Combine like terms: (4x ⸺ 8)

This method ensures expressions are simplified correctly and efficiently, enhancing problem-solving skills in algebra.

Example Problems

Practice simplifying expressions by applying the distributive property and combining like terms. Example problems include:

Problem 1: Simplifying with Like Terms and Distributive Property

Simplify: 6x ⎯ 2(x + 4)

Problem 2: Real-World Application of Both Concepts

Simplify: -9x ⸺ 2x(x ⎯ 4)

Simplify the expression: 6x ⎯ 2(x + 4)

Solution:

1. Apply the distributive property to expand the expression:
   
   6x ⸺ 2(x + 4) = 6x ⎯ 2x ⸺ 8
2. Combine like terms (6x and -2x):
   
   6x ⎯ 2x = 4x
3. Write the simplified expression:
   
   4x ⸺ 8

The final simplified form is 4x ⸺ 8.

A bakery sells a total of 150 loaves of bread per day. They offer two types of loaves: whole wheat and white bread. Let ( w ) represent the number of whole wheat loaves sold, and ( b ) represent the number of white bread loaves sold. The profit from whole wheat loaves is $1.50 each, and the profit from white bread is $1.00 each. If the total profit is $187.50, write and simplify an equation to find the total number of loaves sold.

Solution:

Write the equation: ( 1.50w + 1.00b = 187.50 )

Combine like terms if applicable and simplify:

Since ( w + b = 150 ), substitute ( b = 150 ⎯ w ) into the profit equation:

( 1.50w + 1.00(150 ⎯ w) = 187.50 )

Apply the distributive property:

( 1.50w + 150 ⸺ 1.00w = 187.50 )

Combine like terms:

( 0.50w + 150 = 187.50 )

Solve for ( w ):

( 0.50w = 37.50 ) → ( w = 75 )

Find ( b ): ( b = 150 ⸺ 75 = 75 )

The bakery sells 75 whole wheat and 75 white bread loaves daily.

Practice Exercises

Enhance your skills with worksheets featuring problems that require both the distributive property and combining like terms. Resources like Kuta Software offer challenging exercises to master these concepts effectively.

Recommended Worksheets for Practice

Several worksheets are available to help practice combining like terms and the distributive property. Kuta Software offers comprehensive exercises with answers, ideal for self-assessment. Worksheetplace.com provides free resources with challenging problems, such as simplifying expressions like 9(m + 8) + 11(3m + 4). These worksheets cater to various skill levels, ensuring a smooth learning curve. Additionally, some resources focus on real-world applications, making the concepts more relatable. Many worksheets are designed to first apply the distributive property and then combine like terms, reinforcing the proper sequence of operations. These exercises are perfect for students seeking to master algebraic manipulation and prepare for advanced mathematics. Regular practice with these worksheets will enhance problem-solving speed and accuracy, building a strong foundation in algebraic principles.

Answer Key

An answer key is an essential resource for verifying solutions to worksheet problems. It provides correct answers and step-by-step solutions, enabling students to check their work and understand where they may have gone wrong. For example, in problems involving the distributive property and combining like terms, the answer key ensures clarity in complex steps. Popular worksheets, such as those from Kuta Software, include detailed answer keys that match each problem. These keys not only list the final answers but also break down the process, making it easier for students to follow. For instance, problems like 6x ⸺ 2(x + 4) or -9(1 ⸺ 3s) are solved systematically, demonstrating how to apply the distributive property first and then combine like terms. Using an answer key helps students identify patterns and improve their problem-solving skills. It also serves as a valuable tool for teachers to assess progress and provide targeted feedback.

Mastering the combination of like terms and the distributive property is a fundamental step in algebraic problem-solving. These skills are essential for simplifying expressions and solving equations accurately. By practicing with worksheets and applying the concepts systematically, students can build confidence and fluency. The ability to first apply the distributive property and then combine like terms is a cornerstone of algebra, preparing learners for more complex mathematical challenges. Regular practice and review of these concepts ensure long-term understanding and proficiency. Worksheets, such as those from Kuta Software, provide structured exercises that cater to different skill levels, making them invaluable for both students and educators. Ultimately, these skills are not only crucial for academic success but also for real-world applications in fields like science, engineering, and finance. Keep practicing to achieve mastery!