Factoring Out the Common Term

Oct 13, 2025 by windailys.com 43 views

Introduction

In algebra, simplifying expressions is a crucial skill that helps us solve equations and inequalities. In this article, we will focus on simplifying a given expression using basic algebraic operations. We will break down the expression into smaller parts, combine like terms, and finally arrive at the simplified form.

The Given Expression

The given expression is: ${$ (4k - 6) + (6x + 1)$}$

Our goal is to simplify this expression by combining like terms and eliminating any unnecessary operations.

Step 1: Distribute the Negative Sign

The first step is to distribute the negative sign to the terms inside the parentheses. This will help us combine like terms later on.

${$ (4k - 6) + (6x + 1) = 4k - 6 + 6x + 1$}$

Step 2: Combine Like Terms

Now that we have distributed the negative sign, 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 x: 6x and 1. We can combine these terms by adding their coefficients.

${ $4k - 6 + 6x + 1 = 4k - 6 + 6x + 1 = 4k + 6x - 5\$}$

Step 3: Simplify the Expression

Now that we have combined like terms, we can simplify the expression by eliminating any unnecessary operations.

In this case, we can eliminate the negative sign in front of the 6 by changing the sign of the 5.

${ $4k + 6x - 5 = 4k + 6x - 5\$}$

The Final Answer

After simplifying the expression, we arrive at the final answer:

${ $4k + 6x - 5\$}$

Comparison with Options

Now that we have simplified the expression, let's compare it with the given options.

Option A: ${ $5x + 6\$}$

Option B: ${ $9x - 5\$}$

Option C: ${$ x - 5$}$

Option D: ${$ -x - 1$}$

As we can see, none of the options match our simplified expression. However, we can try to match the expression with the options by rearranging the terms.

Rearranging the Terms

Let's try to rearrange the terms in the simplified expression to match one of the options.

${ $4k + 6x - 5 = 6x + 4k - 5\$}$

Now, let's compare this expression with the options.

Option A: ${ $5x + 6\$}$

Option B: ${ $9x - 5\$}$

Option C: ${$ x - 5$}$

Option D: ${$ -x - 1$}$

As we can see, none of the options match our rearranged expression. However, we can try to match the expression with the options by factoring out the common term.

Factoring Out the Common Term

Let's try to factor out the common term in the simplified expression.

${ $4k + 6x - 5 = 2(2k + 3x) - 5\$}$

Now, let's compare this expression with the options.

Option A: ${ $5x + 6\$}$

Option B: ${ $9x - 5\$}$

Option C: ${$ x - 5$}$

Option D: ${$ -x - 1$}$

As we can see, none of the options match our factored expression. However, we can try to match the expression with the options by rearranging the terms again.

Rearranging the Terms Again

Let's try to rearrange the terms in the factored expression to match one of the options.

Q&A: Simplifying Expressions

Q: What is simplifying an expression?

A: Simplifying an expression is the process of combining like terms and eliminating any unnecessary operations to arrive at a simpler form.

Q: Why is simplifying expressions important?

A: Simplifying expressions is important because it helps us solve equations and inequalities more efficiently. By simplifying expressions, we can identify patterns and relationships between variables, which can lead to a deeper understanding of the underlying math.

Q: What are like terms?

A: Like terms are terms that have the same variable raised to the same power. For example, 2x and 4x are like terms because they both have the variable x raised to the power of 1.

Q: How do I combine like terms?

A: To combine like terms, we add or subtract their coefficients. For example, 2x + 4x = 6x, and 3x - 2x = x.

Q: What is the distributive property?

A: The distributive property is a rule that allows us to distribute a coefficient to multiple terms. For example, 2(3x + 4) = 6x + 8.

Q: How do I simplify an expression using the distributive property?

A: To simplify an expression using the distributive property, we distribute the coefficient to each term inside the parentheses. For example, 2(3x + 4) = 6x + 8.

Q: What is factoring?

A: Factoring is the process of expressing an expression as a product of simpler expressions. For example, 6x can be factored as 2(3x).

Q: How do I factor an expression?

A: To factor an expression, we look for common factors among the terms. For example, 6x can be factored as 2(3x) because 2 is a common factor of 6 and 3x.

Q: What is the difference between simplifying and factoring?

A: Simplifying and factoring are two different processes. Simplifying involves combining like terms and eliminating unnecessary operations, while factoring involves expressing an expression as a product of simpler expressions.

Q: Can you give me an example of simplifying an expression?

A: Let's simplify the expression: [ $(4k - 6) + (6x + 1)\$}$

Using the distributive property, we can rewrite the expression as:

${ $4k - 6 + 6x + 1 = 4k + 6x - 5\$}$

This is the simplified form of the expression.

Q: Can you give me an example of factoring an expression?

A: Let's factor the expression: ${ $6x\$}$

We can factor the expression as:

${ $6x = 2(3x)\$}$

This is the factored form of the expression.

Conclusion

Simplifying expressions is an important skill in algebra that helps us solve equations and inequalities more efficiently. By combining like terms and eliminating unnecessary operations, we can arrive at a simpler form of the expression. Factoring is another important skill that involves expressing an expression as a product of simpler expressions. By understanding the difference between simplifying and factoring, we can tackle more complex math problems with confidence.