The Rule of Subtracting Coefficients

Oct 13, 2025 by miodeya.com 93 views

Understanding the Basics of Algebraic Expressions

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill to master. In this article, we will explore the facts that can be applied to simplify the given expression: $(-5x - 3y) - (-x - 3z)$ . We will examine three options and discuss their relevance to simplifying algebraic expressions.

Option 1: To Subtract Like Terms, Subtract the Coefficients, Not the Variables

When simplifying algebraic expressions, it is essential to understand the concept of like terms. Like terms are terms that contain the same variable, raised to the same power. In the given expression, we have two sets of like terms: $-5x$ and $-x$ , and $-3y$ and $-3z$ . To simplify the expression, we need to subtract the coefficients of these like terms.

The Rule of Subtracting Coefficients

The rule states that when subtracting like terms, we subtract the coefficients, not the variables. This means that we will subtract the numerical values associated with the variables, not the variables themselves. For example, when subtracting $-5x$ and $-x$ , we will subtract the coefficients $-5$ and $-1$ , resulting in $-4x$ .

Applying the Rule to the Given Expression

Let's apply the rule to the given expression: $(-5x - 3y) - (-x - 3z)$ . We can start by simplifying the first set of like terms: $-5x$ and $-x$ . Subtracting the coefficients, we get $-4x$ . Next, we can simplify the second set of like terms: $-3y$ and $-3z$ . Subtracting the coefficients, we get $0$ .

Simplifying the Expression

Now that we have simplified the like terms, we can rewrite the expression as: $-4x - 3y + 3z$ . This is the simplified form of the given expression.

Option 2: Like Terms are Terms that Contain the Same Variable, Raised to the Same Power

This option is a fundamental concept in algebra, and it is essential to understand the definition of like terms. Like terms are terms that contain the same variable, raised to the same power. In the given expression, we have two sets of like terms: $-5x$ and $-x$ , and $-3y$ and $-3z$ .

The Definition of Like Terms

The definition of like terms states that two terms are like terms if they contain the same variable, raised to the same power. For example, $2x$ and $-3x$ are like terms because they contain the same variable $x$ , raised to the same power.

Applying the Definition to the Given Expression

Let's apply the definition to the given expression: $(-5x - 3y) - (-x - 3z)$ . We can start by identifying the like terms: $-5x$ and $-x$ , and $-3y$ and $-3z$ . Since these terms contain the same variable, raised to the same power, we can simplify them using the rule of subtracting coefficients.

Simplifying the Expression

As we discussed earlier, we can simplify the like terms by subtracting the coefficients. This results in the simplified form of the expression: $-4x - 3y + 3z$ .

Option 3: Distributive Property

The distributive property is a fundamental concept in algebra, and it can be applied to simplify algebraic expressions. The distributive property states that for any real numbers $a$ , $b$ , and $c$ , the following equation holds: $a(b + c) = ab + ac$ .

Applying the Distributive Property to the Given Expression

Let's apply the distributive property to the given expression: $(-5x - 3y) - (-x - 3z)$ . We can start by distributing the negative sign to the terms inside the parentheses: $-5x - 3y + x + 3z$ .

Simplifying the Expression

Now that we have applied the distributive property, we can simplify the expression by combining like terms. We can start by combining the $x$ terms: $-5x + x = -4x$ . Next, we can combine the $y$ terms: $-3y + 0 = -3y$ . Finally, we can combine the $z$ terms: $3z + 0 = 3z$ .

Conclusion

In conclusion, we have explored three options for simplifying the given expression: $(-5x - 3y) - (-x - 3z)$ . We have discussed the rule of subtracting coefficients, the definition of like terms, and the distributive property. By applying these concepts, we have simplified the expression to its final form: $-4x - 3y + 3z$ . This example demonstrates the importance of understanding the basics of algebraic expressions and applying the relevant rules and concepts to simplify complex expressions.

Final Answer

Frequently Asked Questions

In this article, we will address some of the most common questions related to simplifying algebraic expressions. Whether you are a student, teacher, or simply someone looking to brush up on their math skills, this guide is for you.

Q: What is the difference between like terms and unlike terms?

A: Like terms are terms that contain the same variable, raised to the same power. Unlike terms are terms that contain different variables or variables raised to different powers.

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

A: To identify like terms, look for terms that contain the same variable, raised to the same power. For example, $2x$ and $-3x$ are like terms because they contain the same variable $x$ , raised to the same power.

Q: What is the rule for subtracting like terms?

A: The rule for subtracting like terms states that when subtracting like terms, we subtract the coefficients, not the variables. This means that we will subtract the numerical values associated with the variables, not the variables themselves.

Q: Can you give an example of subtracting like terms?

A: Let's consider the expression $(-5x - 3y) - (-x - 3z)$ . To simplify this expression, we need to subtract the coefficients of the like terms. We can start by simplifying the first set of like terms: $-5x$ and $-x$ . Subtracting the coefficients, we get $-4x$ . Next, we can simplify the second set of like terms: $-3y$ and $-3z$ . Subtracting the coefficients, we get $0$ .

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 $c$ , the following equation holds: $a(b + c) = ab + ac$ . This property can be used to simplify algebraic expressions by distributing the coefficients to the terms inside the parentheses.

Q: Can you give an example of using the distributive property to simplify an algebraic expression?

A: Let's consider the expression $(-5x - 3y) - (-x - 3z)$ . We can start by distributing the negative sign to the terms inside the parentheses: $-5x - 3y + x + 3z$ . Next, we can simplify the expression by combining like terms. We can start by combining the $x$ terms: $-5x + x = -4x$ . Next, we can combine the $y$ terms: $-3y + 0 = -3y$ . Finally, we can combine the $z$ terms: $3z + 0 = 3z$ .

Q: What is the final form of the expression $(-5x - 3y) - (-x - 3z)$ ?

A: The final form of the expression $(-5x - 3y) - (-x - 3z)$ is $-4x - 3y + 3z$ .

Q: How do I know when to use the distributive property and when to use the rule for subtracting like terms?

A: The distributive property is used when you have an expression with parentheses and you need to distribute the coefficients to the terms inside the parentheses. The rule for subtracting like terms is used when you have like terms in an expression and you need to simplify them.

Q: Can you give me some tips for simplifying algebraic expressions?

A: Here are some tips for simplifying algebraic expressions:

Identify like terms and combine them using the rule for subtracting like terms.
Use the distributive property to simplify expressions with parentheses.
Combine constants and variables separately.
Simplify expressions by canceling out common factors.

Conclusion

In conclusion, simplifying algebraic expressions is a crucial skill to master in mathematics. By understanding the basics of algebraic expressions, applying the relevant rules and concepts, and practicing with examples, you can become proficient in simplifying complex expressions. Remember to identify like terms, use the distributive property, and combine constants and variables separately to simplify expressions. With practice and patience, you will become a master of simplifying algebraic expressions.