What Is The Simplified Expression For The Expression Below?${ 4(3x - 2) + 6x(2 - 1) }$A. { 18x - 8 $}$B. { 18x - 7 $}$C. { 24x - 14 $}$

Understanding the Problem

The given problem involves simplifying an algebraic expression using the distributive property and combining like terms. The expression to be simplified is $4(3x - 2) + 6x(2 - 1)$. Our goal is to simplify this expression and choose the correct answer from the options provided.

Step 1: Apply the Distributive Property

To simplify the given expression, we need to apply the distributive property, which states that for any real numbers $a$, $b$, and $c$, $a(b + c) = ab + ac$. We can apply this property to both terms in the expression.

4(3x - 2) + 6x(2 - 1) = 4(3x) - 4(2) + 6x(2) - 6x(1)

Step 2: Simplify Each Term

Now, we can simplify each term by multiplying the numbers outside the parentheses with the terms inside.

4(3x) - 4(2) + 6x(2) - 6x(1) = 12x - 8 + 12x - 6x

Step 3: Combine Like Terms

The next step is to combine like terms, which involves adding or subtracting terms that have the same variable and exponent. In this case, we can combine the terms with the variable $x$.

12x - 8 + 12x - 6x = 18x - 8

Conclusion

After simplifying the given expression using the distributive property and combining like terms, we arrive at the simplified expression $18x - 8$. This expression matches option A, which is the correct answer.

Answer

The simplified expression for the given algebraic expression is $18x - 8$. Therefore, the correct answer is:

A. $18x - 8$

Why is this the Correct Answer?

The correct answer is $18x - 8$ because it is the result of simplifying the given expression using the distributive property and combining like terms. This answer is obtained by applying the rules of algebra and simplifying the expression step by step.

What is the Importance of Simplifying Algebraic Expressions?

Simplifying algebraic expressions is an essential skill in mathematics, as it helps to:

• Reduce complexity: Simplifying expressions makes them easier to work with and understand.
• Reveal underlying patterns: Simplifying expressions can help reveal underlying patterns and relationships between variables.
• Make calculations easier: Simplified expressions can make calculations easier and more efficient.

Common Mistakes to Avoid When Simplifying Algebraic Expressions

When simplifying algebraic expressions, it's essential to avoid common mistakes such as:

• Forgetting to distribute: Failing to distribute the numbers outside the parentheses to the terms inside.
• Not combining like terms: Failing to combine terms with the same variable and exponent.
• Making arithmetic errors: Making errors when performing arithmetic operations such as addition, subtraction, multiplication, and division.

Tips for Simplifying Algebraic Expressions

To simplify algebraic expressions effectively, follow these tips:

• Apply the distributive property: Use the distributive property to simplify expressions by distributing numbers outside the parentheses to the terms inside.
• Combine like terms: Combine terms with the same variable and exponent to simplify the expression.
• Check your work: Double-check your work to ensure that you have simplified the expression correctly.

Conclusion

Q: What is the distributive property, and how is it used in simplifying algebraic expressions?

A: The distributive property is a fundamental concept in algebra that states that for any real numbers $a$, $b$, and $c$, $a(b + c) = ab + ac$. This property is used to simplify expressions by distributing numbers outside the parentheses to the terms inside.

Q: How do I apply the distributive property to simplify an algebraic expression?

A: To apply the distributive property, simply multiply the numbers outside the parentheses to the terms inside. For example, if you have the expression $4(3x - 2)$, you would multiply 4 to both terms inside the parentheses, resulting in $12x - 8$.

Q: What is the difference between combining like terms and simplifying an algebraic expression?

A: Combining like terms involves adding or subtracting terms that have the same variable and exponent. Simplifying an algebraic expression, on the other hand, involves using the distributive property and combining like terms to reduce the complexity of the expression.

Q: How do I know which terms to combine when simplifying an algebraic expression?

A: To determine which terms to combine, look for terms that have the same variable and exponent. For example, if you have the expression $12x + 6x$, you can combine the two terms by adding their coefficients, resulting in $18x$.

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

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

• Forgetting to distribute numbers outside the parentheses to the terms inside.
• Not combining like terms.
• Making arithmetic errors when performing operations such as addition, subtraction, multiplication, and division.

Q: How can I check my work when simplifying an algebraic expression?

A: To check your work, simply re-evaluate the expression using the distributive property and combining like terms. You can also use a calculator or a computer algebra system to verify your answer.

Q: What are some tips for simplifying algebraic expressions?

A: Some tips for simplifying algebraic expressions include:

• Apply the distributive property to simplify expressions by distributing numbers outside the parentheses to the terms inside.
• Combine like terms to reduce the complexity of the expression.
• Check your work to ensure that you have simplified the expression correctly.

Q: Why is it important to simplify algebraic expressions?

A: Simplifying algebraic expressions is important because it helps to:

• Reduce complexity: Simplifying expressions makes them easier to work with and understand.
• Reveal underlying patterns: Simplifying expressions can help reveal underlying patterns and relationships between variables.
• Make calculations easier: Simplified expressions can make calculations easier and more efficient.

Q: Can you provide examples of algebraic expressions that can be simplified using the distributive property and combining like terms?

A: Yes, here are some examples of algebraic expressions that can be simplified using the distributive property and combining like terms:

• $4(3x - 2) + 6x(2 - 1)$
• $12x + 6x$
• $3(2x + 5) - 2(3x - 2)$

These expressions can be simplified using the distributive property and combining like terms to arrive at the simplified expression.

Conclusion

Simplifying algebraic expressions is an essential skill in mathematics that helps to reduce complexity, reveal underlying patterns, and make calculations easier. By applying the distributive property, combining like terms, and checking your work, you can simplify algebraic expressions effectively. Remember to avoid common mistakes and follow the tips provided to simplify expressions with confidence.