Hey guys! Let's dive into the fascinating world of algebraic expressions and break down the expression (6t + 19z)(7y - 5). This might look intimidating at first, but trust me, with a systematic approach, we can easily figure out what's going on. We'll explore different ways to describe this expression and learn some key mathematical concepts along the way. So, buckle up and let's get started!
Understanding the Basics: What is an Algebraic Expression?
First off, let's define what an algebraic expression actually is. Think of it as a mathematical phrase that combines numbers, variables, and operation symbols (+, -, ×, ÷). Our expression, (6t + 19z)(7y - 5), perfectly fits this definition. We've got variables like 't', 'z', and 'y', constants like 6, 19, 7, and 5, and operations like addition, subtraction, and multiplication implied by the parentheses.
The real magic of algebra comes from the fact that these variables can represent unknown values. This allows us to create general formulas and relationships that apply to a wide range of situations. For instance, 't' could represent time, 'z' could represent the number of zebras (why not?), and 'y' could represent the yield of a crop. The possibilities are endless!
Now, let's talk about the components of our expression. We have two main parts enclosed in parentheses: (6t + 19z) and (7y - 5). Each of these parts is a binomial, which simply means an expression with two terms. The first binomial, (6t + 19z), contains the terms 6t and 19z. These terms are connected by addition. Similarly, the second binomial, (7y - 5), has the terms 7y and -5, connected by subtraction. Understanding these building blocks is crucial for describing the entire expression accurately.
But wait, there's more! The fact that these two binomials are placed next to each other with parentheses implies multiplication. This is a fundamental concept in algebra. So, what we're really looking at is the product of two binomials. This is a very common type of expression in algebra, and knowing how to handle it is a valuable skill. We'll delve deeper into what this multiplication entails later, but for now, just remember that (6t + 19z)(7y - 5) represents the multiplication of two binomial expressions. Describing the expression in this way is a crucial first step in understanding its behavior and how to work with it. It sets the stage for further analysis, simplification, and manipulation. Without this fundamental understanding, we'd be lost in a sea of variables and numbers!
Describing the Expression: Key Characteristics
Okay, now that we've got the basics down, let's focus on how to best describe the expression (6t + 19z)(7y - 5). There are several ways we can approach this, depending on what we want to emphasize. We could describe it in terms of its structure, its degree, or the operations involved. Let's explore some key characteristics.
First and foremost, as we discussed earlier, this expression is the product of two binomials. This is a crucial observation. Each binomial contains two terms: (6t + 19z) and (7y - 5). Recognizing this structure immediately tells us a lot about how the expression will behave when we manipulate it. For example, we know that to fully expand this expression, we'll need to use the distributive property (or the FOIL method, which is a special case of the distributive property).
Another important characteristic is the degree of the expression. The degree refers to the highest power of the variables in the expression. To determine the degree, we first need to consider the degree of each term. In this case, each term within the binomials has a degree of 1. For instance, 6t has a degree of 1 because 't' is raised to the power of 1 (even though it's not explicitly written). Similarly, 19z and 7y also have a degree of 1. The constant term -5 has a degree of 0 (since it can be thought of as -5 times a variable raised to the power of 0).
Now, when we multiply the two binomials, we're essentially multiplying terms. When you multiply variables with exponents, you add the exponents. So, when we multiply a term with 't' by a term with 'y', we'll get a term with 'ty', which has a degree of 2 (1 + 1 = 2). This means that the overall degree of the expression (6t + 19z)(7y - 5) is 2. We call this a quadratic expression because the highest degree is 2. Understanding the degree helps us classify the expression and predict its behavior, especially when we're dealing with equations or functions derived from this expression.
Furthermore, we can describe the expression in terms of the operations involved. The primary operation is multiplication, as the two binomials are being multiplied together. Within each binomial, we have addition and subtraction. This combination of operations is quite common in algebraic expressions and equations. Recognizing these operations helps us determine the order in which we need to perform them (remember PEMDAS/BODMAS?). Describing the expression using these characteristics gives us a solid foundation for further analysis and manipulation. We know its structure, its degree, and the operations it involves, which are all key pieces of information for working with it effectively.
Expanding the Expression: A Step-by-Step Guide
So far, we've described the expression (6t + 19z)(7y - 5) in terms of its structure and key characteristics. But to truly understand it, let's expand it! Expanding the expression means multiplying out the binomials to get a simplified form. We can do this using the distributive property (or the FOIL method, which is a shortcut for binomial multiplication). Let's break it down step by step.
The distributive property states that a(b + c) = ab + ac. In our case, we need to distribute each term in the first binomial to each term in the second binomial. Think of it like this: we're going to multiply 6t by both 7y and -5, and then we're going to multiply 19z by both 7y and -5.
Let's start with the first term, 6t. We multiply 6t by 7y to get 42ty. Then, we multiply 6t by -5 to get -30t. So, the first part of our expanded expression is 42ty - 30t.
Next, let's move on to the second term, 19z. We multiply 19z by 7y to get 133zy (or 133yz, since the order of variables doesn't matter). Then, we multiply 19z by -5 to get -95z. So, the second part of our expanded expression is 133yz - 95z.
Now, we combine these parts together: 42ty - 30t + 133yz - 95z. This is the expanded form of the expression (6t + 19z)(7y - 5). Notice that we have four terms in the expanded form, which is what we'd expect when multiplying two binomials.
We should also check if we can simplify the expression further. In this case, there are no like terms to combine. Like terms are terms that have the same variables raised to the same powers. For example, 3x^2 and -5x^2 are like terms, but 3x^2 and 2x are not. In our expanded expression, we have terms with ty, t, yz, and z. Since none of these terms have the same variable combinations, we cannot simplify further.
So, the final expanded form of the expression is 42ty - 30t + 133yz - 95z. This form is often more useful for certain operations, such as evaluating the expression for specific values of the variables or solving equations. Expanding the expression is a powerful tool for understanding its behavior and working with it in different contexts.
Why is This Important? Applications and Implications
You might be wondering,