Simplify The Expression: $4x(2y) + 3y(2-x$\]A. $5xy + 6y$ B. $8xy + 6y - 3x$ C. $11xy + 6y$ D. $8xy + 6y - X$

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Understanding the Expression

The given expression is $4x(2y) + 3y(2-x)$. To simplify this expression, we need to apply the distributive property and combine like terms. The distributive property states that for any real numbers $a$, $b$, and $c$, $a(b+c) = ab + ac$. We will use this property to expand the given expression.

Step 1: Apply the Distributive Property

To simplify the expression, we will first apply the distributive property to each term. This means that we will multiply each term inside the parentheses by the term outside the parentheses.

$4x(2y) + 3y(2-x)$

Using the distributive property, we get:

$8xy + 3y(2-x)$

Now, we will apply the distributive property to the second term:

$8xy + 6y - 3xy$

Step 2: Combine Like Terms

Now that we have applied the distributive property, we can combine like terms. Like terms are terms that have the same variable raised to the same power. In this case, we have two terms with the variable $xy$ and two terms with the variable $y$.

$8xy + 6y - 3xy$

Combining the like terms, we get:

$5xy + 6y$

Step 3: Check the Answer

Now that we have simplified the expression, we can check our answer by plugging in some values for $x$ and $y$. Let's say $x = 1$ and $y = 2$. Plugging these values into the original expression, we get:

$4(1)(2(2)) + 3(2)(2-1)$

Simplifying this expression, we get:

$16 + 6 = 22$

Now, let's plug these values into the simplified expression:

$5(1)(2) + 6(2)$

Simplifying this expression, we get:

$10 + 12 = 22$

As we can see, the simplified expression $5xy + 6y$ is equal to the original expression when $x = 1$ and $y = 2$. Therefore, our answer is correct.

Conclusion

In this article, we simplified the expression $4x(2y) + 3y(2-x)$ using the distributive property and combining like terms. We found that the simplified expression is $5xy + 6y$. We also checked our answer by plugging in some values for $x$ and $y$ and found that the simplified expression is equal to the original expression.

Final Answer

The final answer is $5xy + 6y$.

Comparison of Options

Let's compare our answer with the options given:

A. $5xy + 6y$ B. $8xy + 6y - 3x$ C. $11xy + 6y$ D. $8xy + 6y - x$

As we can see, our answer matches option A.

Key Takeaways

• The distributive property states that for any real numbers $a$, $b$, and $c$, $a(b+c) = ab + ac$.
• To simplify an expression, we can apply the distributive property and combine like terms.
• We can check our answer by plugging in some values for the variables.

Common Mistakes

• Not applying the distributive property correctly.
• Not combining like terms correctly.
• Not checking the answer by plugging in some values for the variables.

Real-World Applications

• Simplifying expressions is an important skill in mathematics and is used in many real-world applications, such as physics, engineering, and economics.
• Understanding the distributive property and combining like terms is essential in solving problems in these fields.

Practice Problems

• Simplify the expression: $3x(2y) + 2y(3-x)$
• Simplify the expression: $2x(3y) + 4y(2-x)$

Conclusion

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Q: What is the distributive property?

A: The distributive property is a mathematical concept that states that for any real numbers $a$, $b$, and $c$, $a(b+c) = ab + ac$. This means that we can multiply a single term by two or more terms inside parentheses.

Q: How do I apply the distributive property?

A: To apply the distributive property, we need to multiply each term inside the parentheses by the term outside the parentheses. For example, if we have the expression $4x(2y)$, we would multiply $4x$ by $2y$ to get $8xy$.

Q: What are like terms?

A: Like terms are terms that have the same variable raised to the same power. For example, $2xy$ and $3xy$ are like terms because they both have the variable $xy$ raised to the power of 1.

Q: How do I combine like terms?

A: To combine like terms, we need to add or subtract the coefficients of the like terms. For example, if we have the expression $2xy + 3xy$, we would combine the like terms to get $5xy$.

Q: Why is it important to simplify expressions?

A: Simplifying expressions is important because it helps us to understand the underlying structure of the expression and to make it easier to work with. Simplifying expressions can also help us to identify patterns and relationships between variables.

Q: Can you give an example of a real-world application of simplifying expressions?

A: Yes, simplifying expressions is an important skill in many real-world applications, such as physics, engineering, and economics. For example, in physics, we may need to simplify expressions to describe the motion of objects or to calculate forces and energies.

Q: How do I check my answer when simplifying an expression?

A: To check your answer when simplifying an expression, you can plug in some values for the variables and see if the simplified expression is equal to the original expression. This can help you to verify that your answer is correct.

Q: What are some common mistakes to avoid when simplifying expressions?

A: Some common mistakes to avoid when simplifying expressions include:

• Not applying the distributive property correctly
• Not combining like terms correctly
• Not checking the answer by plugging in some values for the variables

Q: Can you give some practice problems for simplifying expressions?

A: Yes, here are some practice problems for simplifying expressions:

• Simplify the expression: $3x(2y) + 2y(3-x)$
• Simplify the expression: $2x(3y) + 4y(2-x)$
• Simplify the expression: $5x(2y) + 3y(4-2x)$

Conclusion

In this article, we answered some common questions about simplifying expressions, including how to apply the distributive property, how to combine like terms, and how to check your answer. We also provided some practice problems for simplifying expressions and discussed some common mistakes to avoid.