Distributive Property: Solving 2 × (8-5) Step-by-Step

The distributive property is a fundamental concept in mathematics that allows us to simplify expressions involving multiplication and addition or subtraction. Understanding and applying the distributive property is crucial for solving algebraic equations and simplifying complex mathematical problems. In this article, we will delve into the distributive property, explain its principles, and demonstrate how to use it effectively. We will specifically focus on solving the expression $2 imes (8-5) = (\square imes 8) - (\square imes 5)$ using the distributive property. By the end of this guide, you'll have a solid grasp of how to apply this property to similar problems and enhance your mathematical skills.

What is the Distributive Property?

The distributive property states that multiplying a single term by a sum or difference inside parentheses is the same as multiplying the term by each number within the parentheses individually and then adding or subtracting the products. Mathematically, it can be expressed as:

• a × (b + c) = (a × b) + (a × c)
• a × (b - c) = (a × b) - (a × c)

Here, 'a' is distributed over both 'b' and 'c'. This property is incredibly useful for simplifying expressions and solving equations. Let's break down the core components of the distributive property to understand it better:

• Term outside the parentheses: This is the number or variable that will be multiplied by each term inside the parentheses. In our example, this is '2'.
• Parentheses: These enclose the sum or difference that needs to be distributed. In our case, the parentheses contain '(8 - 5)'.
• Terms inside the parentheses: These are the numbers or variables that will be multiplied by the term outside the parentheses. Here, the terms are '8' and '5'.

Understanding these components is essential for correctly applying the distributive property. Now, let's move on to solving our specific problem.

Applying the Distributive Property to 2 × (8-5)

Now, let's apply the distributive property to solve the expression $2 imes (8-5) = (\square imes 8) - (\square imes 5)$. This expression perfectly illustrates how the distributive property works with subtraction. Here’s how we can break it down step by step:

1. Identify the terms:
   • The term outside the parentheses is 2.
   • The terms inside the parentheses are 8 and -5 (note the subtraction).
2. Distribute the 2:
   • Multiply 2 by 8: $2 imes 8$
   • Multiply 2 by -5: $2 imes -5$
3. Rewrite the expression:
   • Using the distributive property, we rewrite the expression as $(2 imes 8) - (2 imes 5)$.
4. Fill in the blanks:
   • Comparing this with the original form $(□ imes 8) - (□ imes 5)$, we can see that the number that fills both blanks is 2.

So, the filled-in expression is $(2 imes 8) - (2 imes 5)$. This step-by-step approach ensures that we correctly apply the distributive property and arrive at the accurate expression. By breaking down the problem into manageable steps, it becomes easier to understand and solve.

Step-by-Step Solution: Filling in the Blanks

To further clarify how we arrived at the solution, let's go through each step in detail. Our initial expression is $2 imes (8-5) = (\square imes 8) - (\square imes 5)$. We want to find the number that fits into the blanks.

1. Start with the distributive property:
   • The distributive property tells us that $a imes (b - c) = (a imes b) - (a imes c)$.
2. Apply the property to our expression:
   • In our case, $a = 2$, $b = 8$, and $c = 5$.
   • So, we have $2 imes (8 - 5)$.
3. Distribute the 2:
   • Multiply 2 by 8: $2 imes 8 = 16$
   • Multiply 2 by 5: $2 imes 5 = 10$
4. Rewrite the expression using the distributed terms:
   • $(2 imes 8) - (2 imes 5)$
5. Fill in the blanks:
   • Comparing $(2 imes 8) - (2 imes 5)$ with $(□ imes 8) - (□ imes 5)$, we can clearly see that the number 2 fits into both blanks.

Thus, the completed expression is $(2 imes 8) - (2 imes 5)$. This detailed breakdown provides a clear understanding of each step involved in applying the distributive property. By following these steps, you can confidently solve similar problems.

Simplifying and Evaluating the Expression

After applying the distributive property and filling in the blanks, we have the expression $(2 imes 8) - (2 imes 5)$. Now, let's simplify and evaluate this expression to find the final numerical answer. This will not only help us confirm our solution but also reinforce our understanding of the distributive property.

1. Perform the multiplications:
   • $2 imes 8 = 16$
   • $2 imes 5 = 10$
2. Substitute the results back into the expression:
   • So, $(2 imes 8) - (2 imes 5)$ becomes $16 - 10$.
3. Perform the subtraction:
   • $16 - 10 = 6$

Therefore, the simplified value of the expression $(2 imes 8) - (2 imes 5)$ is 6. To verify our answer, we can also solve the original expression directly:

1. Solve the expression inside the parentheses:
   • $8 - 5 = 3$
2. Multiply by 2:
   • $2 imes 3 = 6$

Both methods give us the same result, 6, which confirms that our application of the distributive property was correct. This process demonstrates the power and utility of the distributive property in simplifying complex expressions.

Common Mistakes to Avoid

While the distributive property is straightforward, it’s easy to make mistakes if you’re not careful. Being aware of these common pitfalls can help you avoid them and ensure accurate problem-solving. Here are some mistakes to watch out for:

1. Forgetting to distribute to all terms:
   • A common mistake is to distribute the term outside the parentheses to only one term inside. For example, incorrectly distributing in $2 imes (8 - 5)$ might look like $(2 imes 8) - 5$, which is wrong. Remember, the 2 must be multiplied by both 8 and 5.
2. Sign errors:
   • When dealing with subtraction, it's crucial to pay attention to the signs. For instance, in $2 imes (8 - 5)$, the negative sign in front of the 5 must be included when distributing, resulting in $(2 imes 8) - (2 imes 5)$.
3. Incorrect order of operations:
   • Another common mistake is not following the correct order of operations (PEMDAS/BODMAS). Always perform the multiplication before the subtraction in the distributed expression.
4. Misunderstanding the property:
   • Some students may misunderstand the distributive property as simply removing the parentheses without proper multiplication. Always ensure that each term inside the parentheses is multiplied by the term outside.

By being mindful of these common errors, you can improve your accuracy and confidence when using the distributive property.

Practice Problems and Further Learning

To truly master the distributive property, practice is key. Here are some practice problems you can try to reinforce your understanding:

1. $3 imes (4 + 2) = (□ imes 4) + (□ imes 2)$
2. $5 imes (7 - 3) = (□ imes 7) - (□ imes 3)$
3. $4 imes (6 + 1) = (□ imes 6) + (□ imes 1)$
4. $2 imes (9 - 4) = (□ imes 9) - (□ imes 4)$
5. $6 imes (3 + 5) = (□ imes 3) + (□ imes 5)$

Solving these problems will help you become more comfortable with the distributive property and its applications. For further learning, you can explore online resources, textbooks, and educational websites that offer more examples and explanations.

Additional Resources

• Khan Academy: Provides comprehensive lessons and practice exercises on the distributive property.
• Mathway: A useful tool for checking your answers and seeing step-by-step solutions.
• Your school’s math resources: Check if your school offers tutoring or additional materials for math help.

By practicing consistently and utilizing available resources, you can build a strong foundation in the distributive property and excel in your mathematical studies.

Conclusion

In conclusion, the distributive property is a vital tool in mathematics that simplifies expressions involving multiplication and addition or subtraction. By understanding and applying this property correctly, you can solve a wide range of algebraic problems more efficiently. In this article, we walked through a detailed example of how to use the distributive property to solve $2 imes (8-5) = (\square imes 8) - (\square imes 5)$, highlighting each step and emphasizing common mistakes to avoid. Remember, the key to mastering the distributive property is practice. By working through various problems and utilizing available resources, you can enhance your mathematical skills and build confidence in your problem-solving abilities. Keep practicing, and you'll find that the distributive property becomes second nature!

For more information on the distributive property and related mathematical concepts, visit Khan Academy's Distributive Property Section.