Find The Missing Number In The Distributive Property Equation

Jul 12, 2025 by ADMIN 62 views

What Number Goes in the Boxes to Make the Number Sentence True?

8 \times (4 + -2) = (\square \times 4) + (\square \times -2)

To determine the number that goes in the boxes to make the number sentence true, we need to apply the distributive property of multiplication over addition. This property states that for any numbers a, b, and c, the following equation holds:

a \times (b + c) = (a \times b) + (a \times c)

In our case, the given number sentence is:

8 \times (4 + -2) = (\square \times 4) + (\square \times -2)

Here, we can see that a = 8, b = 4, and c = -2. Comparing this with the distributive property equation, it's clear that the number that should go in the boxes is 8. Let's delve deeper into why this is the case and explore the underlying mathematical principles.

Understanding the Distributive Property

The distributive property is a fundamental concept in algebra that allows us to simplify expressions involving multiplication and addition (or subtraction). It essentially states that multiplying a number by a sum (or difference) is the same as multiplying the number by each term in the sum (or difference) individually and then adding (or subtracting) the products. This property is crucial for expanding expressions, solving equations, and simplifying mathematical problems.

In the given equation, we have:

8 \times (4 + -2) = (\square \times 4) + (\square \times -2)

Applying the distributive property, we multiply 8 by both 4 and -2 separately:

8 \times (4 + -2) = (8 \times 4) + (8 \times -2)

This clearly shows that the number that goes in the boxes is 8. By substituting 8 into the equation, we get:

8 \times (4 + -2) = (8 \times 4) + (8 \times -2)

Now, let's verify if this equation holds true by performing the calculations on both sides.

Verifying the Solution

First, let's simplify the left side of the equation:

8 \times (4 + -2) = 8 \times (4 - 2) = 8 \times 2 = 16

Next, let's simplify the right side of the equation:

(8 \times 4) + (8 \times -2) = 32 + (-16) = 32 - 16 = 16

Since both sides of the equation equal 16, we have verified that our solution is correct. The number 8 indeed makes the number sentence true.

16 = 16

This confirms that the distributive property is correctly applied, and the number 8 is the solution to the problem. Understanding and applying the distributive property is essential for solving a wide range of algebraic problems.

Why the Other Options are Incorrect

Let's briefly discuss why the other options (F. -8, G. -2, H. 4) are incorrect. Substituting each of these values into the boxes would result in an equation that is not true.

F. -8:

8 \times (4 + -2) = (-8 \times 4) + (-8 \times -2)
8 \times 2 = -32 + 16
16 = -16 (Incorrect)

G. -2:

8 \times (4 + -2) = (-2 \times 4) + (-2 \times -2)
8 \times 2 = -8 + 4
16 = -4 (Incorrect)

H. 4:

8 \times (4 + -2) = (4 \times 4) + (4 \times -2)
8 \times 2 = 16 + (-8)
16 = 8 (Incorrect)

As we can see, none of these substitutions result in a true equation, further confirming that J. 8 is the correct answer.

Key Takeaways

The distributive property is a fundamental concept in algebra that allows us to simplify expressions involving multiplication and addition (or subtraction).
The distributive property states that a × (b + c) = (a × b) + (a × c).
To solve the given problem, we applied the distributive property to identify the number that should go in the boxes.
We verified our solution by performing the calculations on both sides of the equation.
Understanding the distributive property is essential for solving a wide range of algebraic problems.

Therefore, the correct answer is J. 8. This exercise highlights the importance of understanding and applying fundamental mathematical properties like the distributive property to solve equations effectively. By mastering these concepts, students can build a strong foundation in algebra and tackle more complex problems with confidence.

Further Applications of the Distributive Property

The distributive property is not just limited to simple arithmetic problems; it's a cornerstone of algebra and is used extensively in various mathematical contexts. Understanding its applications can significantly enhance problem-solving skills. Here are some further applications of the distributive property:

Simplifying Algebraic Expressions: The distributive property is often used to simplify algebraic expressions by removing parentheses. For example:
```
3(x + 2) = 3x + 6
```
Here, the 3 is distributed to both x and 2, simplifying the expression.
Factoring Algebraic Expressions: Factoring is the reverse process of distribution. We use the distributive property to factor out a common factor from an expression. For example:
```
4x + 8 = 4(x + 2)
```
Here, 4 is factored out from both terms.
Solving Equations: The distributive property is crucial for solving equations, especially those involving parentheses. By distributing, we can simplify the equation and isolate the variable. For example:
```
2(x - 3) = 10
2x - 6 = 10
2x = 16
x = 8
```
Polynomial Multiplication: When multiplying polynomials, the distributive property is applied multiple times. For example:
```
(x + 2)(x + 3) = x(x + 3) + 2(x + 3) = x^2 + 3x + 2x + 6 = x^2 + 5x + 6
```
Mental Math: The distributive property can be used as a mental math technique to simplify calculations. For example, to calculate 6 × 102 mentally, you can think of it as 6 × (100 + 2) = (6 × 100) + (6 × 2) = 600 + 12 = 612.

These applications highlight the versatility and importance of the distributive property in mathematics. Mastering this property can significantly improve one's ability to manipulate algebraic expressions, solve equations, and perform mental calculations.

Practice Problems

To reinforce your understanding of the distributive property, consider solving the following practice problems:

Simplify: 5(x + 4)
Expand: -2(3y - 1)
Factor: 9a + 12
Solve for x: 3(x + 2) = 15
Multiply: (a - 1)(a + 4)

By working through these problems, you'll gain confidence in applying the distributive property and develop a deeper understanding of its applications. Remember, practice is key to mastering mathematical concepts.

In conclusion, the distributive property is a fundamental tool in mathematics with wide-ranging applications. The number that goes in the boxes to make the number sentence true in the given problem is 8, and this is a direct application of the distributive property. By understanding and practicing this property, you'll be well-equipped to tackle various algebraic challenges.