Simplifying Expressions Combining Like Terms In 20(-1.5r + 0.75)

In mathematics, simplifying expressions is a fundamental skill. One key technique for simplification is combining like terms. This involves identifying terms within an expression that share the same variable and exponent, and then adding or subtracting their coefficients. By doing this, we can reduce the complexity of an expression and make it easier to work with. This article will focus on how to combine like terms to create an equivalent expression, specifically looking at the example: $20(-1.5r + 0.75)$.

Understanding Like Terms

Before diving into the example, let's first define what like terms are. Like terms are terms that have the same variable raised to the same power. The coefficients (the numbers in front of the variables) can be different, but the variable part must be identical. For example, $3x$ and $-5x$ are like terms because they both have the variable $x$ raised to the power of 1. However, $3x$ and $3x^2$ are not like terms because the variables are raised to different powers.

Similarly, constants (numbers without any variables) are also like terms. For example, 7 and -2 are like terms. We can combine them by simply adding or subtracting them as needed.

In the expression $20(-1.5r + 0.75)$, we have a term with the variable $r$ and a constant term inside the parentheses. The goal is to simplify this expression by distributing the 20 and then combining any like terms that may arise. This process involves the distributive property, which is a cornerstone of simplifying algebraic expressions. The distributive property states that for any numbers $a$, $b$, and $c$, $a(b + c) = ab + ac$. This property allows us to multiply a single term by each term inside a set of parentheses, effectively expanding the expression. Understanding and applying the distributive property is crucial for correctly simplifying expressions and solving equations.

For example, let's take a look at the expression $3(2x + 5)$. To simplify this, we distribute the 3 to both terms inside the parentheses: $3 * 2x + 3 * 5$. This gives us $6x + 15$. Here, $6x$ is a term with the variable $x$, and 15 is a constant term. Since these terms are not alike, we cannot combine them further. This simple illustration highlights how the distributive property works in conjunction with the concept of like terms to simplify expressions. By mastering these basic principles, we can tackle more complex algebraic problems with confidence. Recognizing like terms is the foundation for simplifying expressions, and the distributive property is the tool we use to manipulate and rearrange these terms effectively.

Applying the Distributive Property

The first step in simplifying the expression $20(-1.5r + 0.75)$ is to apply the distributive property. This means we multiply the 20 by each term inside the parentheses:

$20 * (-1.5r) + 20 * (0.75)$

Now, let's perform the multiplication:

$-30r + 15$

So, after distributing the 20, our expression becomes $-30r + 15$. At this stage, it’s important to carefully review the resulting terms to identify any potential like terms. In our expanded expression, $-30r$ is a term that includes the variable $r$, and 15 is a constant term. Remember, like terms are terms that have the same variable raised to the same power. Since $-30r$ has the variable $r$ and 15 is a constant, they are not like terms. This means we cannot combine them any further. The distributive property has allowed us to remove the parentheses and transform the original expression into a sum of individual terms.

This step is crucial in the simplification process because it breaks down the expression into more manageable parts. By distributing the term outside the parentheses, we can clearly see all the individual terms and their coefficients. This makes it easier to identify and combine like terms in the subsequent steps. It’s like taking apart a complex machine to see all its individual components – once we understand the parts, we can put them back together in a simplified way. In the context of algebra, the distributive property is our primary tool for “disassembling” expressions, making them more accessible and easier to manipulate. This process is not just about following a set of rules; it’s about understanding the structure of the expression and how the distributive property helps us to reveal that structure.

Identifying and Combining Like Terms

After applying the distributive property, we have the expression $-30r + 15$. Now, we need to identify if there are any like terms that can be combined. In this case, we have two terms: $-30r$ and $15$.

As we discussed earlier, like terms have the same variable raised to the same power. The term $-30r$ has the variable $r$, while the term $15$ is a constant. Since they do not have the same variable, they are not like terms.

Therefore, we cannot combine these terms any further. The expression $-30r + 15$ is already in its simplest form. Guys, remember that recognizing like terms is a crucial step in simplifying algebraic expressions. It’s like sorting through a pile of mixed objects and grouping together the ones that are similar. In algebra, this means identifying terms that have the same variable and exponent. For instance, $5x$ and $-2x$ are like terms because they both have the variable $x$ raised to the first power. On the other hand, $3x^2$ and $7x$ are not like terms because the variable $x$ is raised to different powers. Similarly, constants like 8 and -4 are considered like terms because they don't have any variables attached to them.

The process of combining like terms involves adding or subtracting their coefficients, while keeping the variable part the same. For example, if we have $5x + (-2x)$, we would add the coefficients 5 and -2 to get 3, and then keep the variable $x$, resulting in $3x$. It's important to pay close attention to the signs of the coefficients when combining like terms. A common mistake is to incorrectly add or subtract negative numbers. By carefully identifying and combining like terms, we can simplify complex expressions into more manageable forms, making them easier to understand and work with.

The Equivalent Expression

Since we cannot combine any further terms, the simplified expression is $-30r + 15$. This expression is equivalent to the original expression, $20(-1.5r + 0.75)$. This means that for any value of $r$, both expressions will produce the same result.

The concept of equivalent expressions is fundamental in algebra. Equivalent expressions are expressions that may look different but have the same value for all possible values of the variables. In other words, they are different forms of the same mathematical statement. Simplifying expressions to their equivalent forms is a core skill in algebra, as it allows us to work with expressions in a more manageable way without changing their fundamental meaning. To ensure that an expression is truly equivalent after simplification, it’s crucial to apply mathematical rules and properties correctly, such as the distributive property and the commutative and associative properties of addition and multiplication.

For example, if we substitute $r = 1$ into both the original expression and the simplified expression, we get:

Original: $20(-1.5(1) + 0.75) = 20(-1.5 + 0.75) = 20(-0.75) = -15$

Simplified: $-30(1) + 15 = -30 + 15 = -15$

As you can see, both expressions yield the same result, -15, demonstrating their equivalence. This principle holds true for any value of $r$, which is the essence of what makes two expressions equivalent. Verifying equivalence by substituting different values for the variables is a useful technique for checking your work and ensuring that you have correctly simplified an expression. It’s like having a built-in error detection system that helps you catch any mistakes you might have made along the way. This not only reinforces your understanding of algebraic principles but also builds confidence in your ability to manipulate expressions accurately.

Conclusion

In summary, to create an equivalent expression from $20(-1.5r + 0.75)$, we first applied the distributive property to get $-30r + 15$. Then, we identified that there were no like terms to combine. Therefore, the equivalent expression is $-30r + 15$. This process of combining like terms is a crucial skill in algebra, allowing us to simplify expressions and solve equations more efficiently. By understanding the concepts of like terms and the distributive property, you can confidently tackle more complex algebraic problems. Keep practicing, guys, and you'll become pros at simplifying expressions!