Distributing the Coefficients

Oct 13, 2025 by miodeya.com 70 views

What is the Simplified Expression for the Given Algebraic Expression?

Understanding the Problem

When dealing with algebraic expressions, simplification is a crucial step to make the expression more manageable and easier to work with. In this problem, we are given an algebraic expression in the form of $7(x-4) - 3(x+5)$ , and we need to simplify it to find the final expression.

Step 1: Distribute the Coefficients

To simplify the given expression, we need to start by distributing the coefficients to the terms inside the parentheses. This means multiplying the coefficient of the first term by the terms inside the first parentheses, and then multiplying the coefficient of the second term by the terms inside the second parentheses.

Distributing the Coefficients

${ 7(x-4) - 3(x+5) }$ ${ = 7x - 28 - 3x - 15 }$

Step 2: Combine Like Terms

Now that we have distributed the coefficients, we can combine like terms to simplify the expression further. Like terms are terms that have the same variable raised to the same power.

Combining Like Terms

${ 7x - 28 - 3x - 15 }$ ${ = (7x - 3x) + (-28 - 15) }$ ${ = 4x - 43 }$

Step 3: Check the Answer Choices

Now that we have simplified the expression, we need to check the answer choices to see which one matches our simplified expression.

Checking the Answer Choices

The answer choices are:

A. $4x - 43$ B. $4x + 1$ C. $4x - 13$ D. $4x - 9$

Our simplified expression is $4x - 43$ , which matches answer choice A.

Conclusion

In conclusion, the simplified expression for the given algebraic expression $7(x-4) - 3(x+5)$ is $4x - 43$ . This is the correct answer choice.

Key Takeaways

When simplifying algebraic expressions, it's essential to distribute the coefficients to the terms inside the parentheses.
Combining like terms is a crucial step in simplifying algebraic expressions.
Checking the answer choices is essential to ensure that the simplified expression matches one of the answer choices.

Final Answer

The final answer is $\boxed{A. 4x - 43}$ .
Frequently Asked Questions (FAQs) about Simplifying Algebraic Expressions

Q: What is the first step in simplifying an algebraic expression?

A: The first step in simplifying an algebraic expression is to distribute the coefficients to the terms inside the parentheses. This means multiplying the coefficient of the first term by the terms inside the first parentheses, and then multiplying the coefficient of the second term by the terms inside the second parentheses.

Q: What are like terms in algebraic expressions?

A: Like terms are terms that have the same variable raised to the same power. For example, $2x$ and $5x$ are like terms because they both have the variable $x$ raised to the power of 1.

Q: How do I combine like terms in an algebraic expression?

A: To combine like terms, you need to add or subtract the coefficients of the like terms. For example, if you have $2x + 5x$ , you can combine the like terms by adding the coefficients: $2x + 5x = 7x$ .

Q: What is the difference between a coefficient and a variable?

A: A coefficient is a number that is multiplied by a variable. For example, in the expression $3x$ , the coefficient is 3 and the variable is $x$ . A variable is a letter or symbol that represents a value that can change.

Q: How do I simplify an algebraic expression with multiple sets of parentheses?

A: To simplify an algebraic expression with multiple sets of parentheses, you need to follow the order of operations (PEMDAS):

Evaluate expressions inside parentheses
Evaluate any exponential expressions
Evaluate any multiplication and division expressions from left to right
Evaluate any addition and subtraction expressions from left to right

Q: What is the purpose of simplifying algebraic expressions?

A: The purpose of simplifying algebraic expressions is to make them easier to work with and to make it easier to solve equations and inequalities. Simplifying algebraic expressions can also help to identify patterns and relationships between variables.

Q: Can you provide an example of a simplified algebraic expression?

A: Yes, here is an example of a simplified algebraic expression:

$2x + 5x - 3x = 4x$

In this example, the like terms $2x$ , $5x$ , and $-3x$ are combined to form the simplified expression $4x$ .

Q: How do I know if an algebraic expression is simplified?

A: An algebraic expression is simplified when there are no like terms that can be combined. In other words, the expression cannot be simplified any further.

Q: Can you provide a real-world example of simplifying algebraic expressions?

A: Yes, here is a real-world example of simplifying algebraic expressions:

Suppose you are a manager of a company and you need to calculate the total cost of producing a certain number of products. The cost of producing each product is given by the expression $2x + 5x - 3x$ , where $x$ is the number of products produced. To simplify this expression, you can combine the like terms to get $4x$ . This makes it easier to calculate the total cost of producing a certain number of products.

Conclusion

In conclusion, simplifying algebraic expressions is an essential skill in mathematics and is used in a wide range of real-world applications. By following the steps outlined in this article, you can simplify algebraic expressions and make them easier to work with.