Simplifying Algebraic Expressions Facts And Techniques

In the realm of mathematics, simplifying expressions is a fundamental skill. It allows us to take complex equations and reduce them to their most basic, manageable forms. This process not only makes expressions easier to understand but also facilitates further calculations and problem-solving. One of the key techniques in simplifying expressions involves identifying and combining like terms. In this article, we will delve into the concept of like terms, explore the rules for combining them, and demonstrate how these principles can be applied to simplify expressions effectively.

Understanding Like Terms

At the heart of simplifying expressions lies the concept of like terms. Like terms are terms that share the same variable, raised to the same power. This means that the variable part of the terms must be identical for them to be considered like terms. For instance, in the expression 3x + 5x - 2y, the terms 3x and 5x are like terms because they both contain the variable x raised to the power of 1. On the other hand, the term -2y is not a like term because it contains the variable y instead of x.

To further illustrate this concept, consider the following examples:

• 4a and -7a are like terms because they both have the variable a raised to the power of 1.
• 2x² and 9x² are like terms because they both have the variable x raised to the power of 2.
• 5xy and -3xy are like terms because they both have the variables x and y raised to the power of 1.
• 6b and 6b² are not like terms because although they share the same variable b, the powers are different (1 and 2).
• 8p and 8q are not like terms because they have different variables (p and q).

Identifying like terms is the first step in simplifying expressions. Once you can recognize like terms, you can then proceed to combine them using the rules of addition and subtraction.

Rules for Combining Like Terms

The rule for combining like terms is straightforward: simply add or subtract the coefficients of the like terms while keeping the variable part the same. The coefficient is the numerical factor that multiplies the variable.

For example, to combine the like terms 3x and 5x, we add their coefficients (3 and 5) and keep the variable part x the same:

3x + 5x = (3 + 5)x = 8x

Similarly, to combine the like terms 7y and -2y, we subtract their coefficients (7 and -2) and keep the variable part y the same:

7y - 2y = (7 - 2)y = 5y

It is important to note that you can only combine like terms. You cannot combine terms that have different variables or different powers of the same variable. For instance, you cannot combine 4x and 3y because they have different variables. Similarly, you cannot combine 2x² and 5x because they have different powers of the variable x.

When simplifying expressions with multiple sets of like terms, it can be helpful to group the like terms together before combining them. This can make the process less prone to errors.

Let's consider an example:

Simplify the expression: 6a + 2b - 4a + 5b

First, we group the like terms together:

(6a - 4a) + (2b + 5b)

Then, we combine the like terms:

2a + 7b

The simplified expression is 2a + 7b.

Applying Like Terms to Simplify Expressions: Step-by-Step

Now, let's walk through a step-by-step example of how to apply the concept of like terms to simplify an expression. We'll use the expression provided in the prompt as our example:

Expression: (-5x - 3y) - (-x - 3z)

Step 1: Distribute the negative sign

The first step is to distribute the negative sign in front of the second set of parentheses. This means multiplying each term inside the parentheses by -1:

-5x - 3y + x + 3z

Step 2: Identify like terms

Next, we identify the like terms in the expression. In this case, we have two terms with the variable x: -5x and +x.

Step 3: Group like terms

We group the like terms together:

(-5x + x) - 3y + 3z

Step 4: Combine like terms

Now, we combine the like terms by adding their coefficients:

(-5 + 1)x - 3y + 3z

-4x - 3y + 3z

Step 5: Write the simplified expression

The simplified expression is -4x - 3y + 3z. This expression is now in its simplest form, as there are no more like terms to combine.

Key Facts for Simplifying Expressions

Based on our exploration of like terms, we can identify three key facts that are crucial for simplifying expressions:

1. To subtract like terms, subtract the coefficients, not the variables. This principle is the cornerstone of combining like terms. The coefficients are the numerical factors that multiply the variables, and it's these coefficients that we add or subtract when simplifying. For instance, in the expression 7x - 3x, we subtract the coefficients (7 and 3) to get 4x. The variable x remains the same because we are combining terms that represent the same quantity.
2. Like terms are terms that contain the same variable, raised to the same power. This definition is essential for identifying which terms can be combined. The variable part of the terms must be identical for them to be considered like terms. For example, 5y² and -2y² are like terms because they both have the variable y raised to the power of 2. However, 5y² and 5y are not like terms because they have different powers of the variable y.
3. The order of terms in an expression does not affect its value. This property, known as the commutative property, allows us to rearrange terms in an expression to group like terms together. For example, the expression 3a + 2b - a + 4b can be rearranged as 3a - a + 2b + 4b to make it easier to identify and combine like terms.

These three facts are fundamental for simplifying expressions effectively and accurately. By understanding and applying these principles, you can confidently tackle a wide range of algebraic problems.

Conclusion

Simplifying expressions is a vital skill in mathematics. By understanding the concept of like terms and following the rules for combining them, you can transform complex expressions into simpler, more manageable forms. Remember to identify like terms, group them together, and then add or subtract their coefficients while keeping the variable part the same. By mastering these techniques, you will be well-equipped to tackle a wide range of algebraic problems and excel in your mathematical endeavors.