Simplifying Expressions: 6p + 7q - 5q + 10

by Jhon Lennon 43 views

Hey guys! Let's dive into the world of simplifying algebraic expressions! Today, we're going to break down how to find the simplified form of the expression: 6p + 7q - 5q + 10. This might seem a little daunting at first, but trust me, it's like putting together a puzzle. We just need to gather all the like terms and combine them. So, grab a pen and paper (or your favorite note-taking app), and let's get started. By the end of this, you'll be a pro at simplifying expressions, ready to tackle more complex equations. The main concept here is to understand what "like terms" are. These are terms that have the same variables raised to the same powers. For example, 7q and -5q are like terms because they both have the variable 'q' to the power of 1 (which we usually don't write). On the other hand, 6p is not a like term with 7q because it has a different variable ('p'), and 10 is also not a like term because it's a constant (a number without a variable).

Simplifying expressions is a fundamental skill in algebra and is used in a wide range of fields, from computer science to economics. The ability to manipulate and simplify algebraic expressions enables you to solve equations more efficiently, understand complex relationships, and build a strong foundation for advanced mathematical concepts. Let's not forget how important mathematics is in everyday life; it's a way of understanding and interpreting the world around us. In this example, we'll learn about combining like terms, which is a foundational concept. Imagine each term as a different type of fruit. You can combine apples with apples but not apples with oranges. Similarly, in algebra, you can combine terms with the same variables but not those with different variables or constants. This concept is the key to solving more complex equations, plotting graphs, and understanding various scientific and engineering principles. The ability to identify like terms and simplify expressions is critical for solving equations, manipulating formulas, and analyzing mathematical models. It's like learning the alphabet before you can read; mastering this skill will unlock many more mathematical doors for you. The basic rules for the order of operations like PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) will be used to correctly solve and provide the final answer. This will make sure that the right terms are simplified and combined correctly. Don't worry, by the time we are done, you'll become more familiar with these operations.

Step-by-Step Simplification

Alright, let's break down the simplification process step by step, so that it's easy to follow. We are going to make it super easy for you to understand, step by step, so even if you're a beginner, you'll be able to follow along. We'll go slow, so you can fully understand each concept. We'll start with the original expression: 6p + 7q - 5q + 10. Our goal here is to group like terms together and combine them. Don’t be intimidated – you got this. The process is straightforward, and with a little practice, you'll be simplifying expressions like a pro. This skill is like a superpower in mathematics; it makes complex problems much easier to handle.

Firstly, we have to recognize the like terms. As a reminder, like terms are terms that contain the same variables raised to the same power. In our expression, we have the following:

  • 6p (This term has the variable 'p')
  • 7q (This term has the variable 'q')
  • -5q (This term also has the variable 'q')
  • 10 (This is a constant, a number without a variable)

From the above, we can identify that 7q and -5q are like terms because they both have the variable 'q'. 6p is a different term with a different variable, and 10 is a constant. The next step is to combine the like terms. We do this by adding or subtracting their coefficients (the numbers in front of the variables). Now that we've found our like terms, let's combine them. We'll start by looking at the 'q' terms: 7q - 5q = 2q. So, we've simplified 7q - 5q to become 2q. Let's rewrite the expression now with the combined terms: 6p + 2q + 10. The constant, 10, is a term, so we will not combine it with any other terms in the expression. Now, we check to see if there are any other terms that we can simplify. In this case, there is not because we have the following variables in the expression, 'p' and 'q', which are not alike, and the constant 10. Thus, we have the simplified expression! Lastly, we will write our simplified expression. After combining like terms, the simplified form of the expression is 6p + 2q + 10. We cannot simplify this expression any further because there are no more like terms to combine.

Combining Like Terms

Let’s zoom in on combining like terms, as this is the heart of the simplification process. Remember, guys, the key to simplifying expressions is to identify and combine like terms. This involves adding or subtracting the coefficients of terms that have the same variables raised to the same powers. For instance, if you have 3x + 2x, you can combine them because they both have 'x'. That gives you 5x. This basic principle applies no matter how complex the expression becomes. Always make sure to pay close attention to the signs (+ or -) in front of the terms. A negative sign can drastically change your calculations. Always remember the rules of addition and subtraction for positive and negative numbers.

When combining like terms, the variables and their exponents do not change; only the coefficients are added or subtracted. For example, if you have 4y² - 2y² + 3y², you combine the coefficients: 4 - 2 + 3 = 5. So, the simplified term is 5y². Now, let's practice and show the actual implementation of this concept by applying the like terms in the original expression: 6p + 7q - 5q + 10. You should only combine 7q - 5q. You will rewrite the original expression by combining those like terms, as: 2q. Rewriting it, you now have: 6p + 2q + 10. It is crucial to remember the order of operations (PEMDAS) to ensure you correctly combine the terms. The expression is now in its simplest form because there are no other like terms to combine. We cannot combine '6p' and '2q' because they have different variables. We cannot combine the constant '10' with any variable terms. By mastering this step, you're not just simplifying expressions; you're building a strong foundation for more complex mathematical concepts. This skill is critical for solving equations, manipulating formulas, and understanding various scientific and engineering principles. The ability to do this correctly will boost your confidence and make you ready to tackle even more complex mathematical problems. Keep practicing; with each problem, you'll become more familiar and comfortable with the process.

The Importance of Order of Operations

Let’s not forget the order of operations! This is often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Why is this important? Because it tells you the correct order in which to perform calculations. Even though we aren’t dealing with exponents, multiplication, division, or parentheses in our specific example, understanding the order of operations is still essential. This will always help you avoid mistakes and ensure that you simplify expressions correctly. So, what is PEMDAS?

  1. Parentheses: Always start by simplifying any expressions inside parentheses or brackets. This is the first step.
  2. Exponents: Next, evaluate any exponents. This means squaring, cubing, or raising numbers to any power.
  3. Multiplication and Division: Perform multiplication and division from left to right.
  4. Addition and Subtraction: Finally, perform addition and subtraction from left to right.

In our expression, 6p + 7q - 5q + 10, we do not need to follow all the steps of PEMDAS because there are no parentheses, exponents, multiplication, or division. We are left with just addition and subtraction. So, we simply combine like terms. The order of operations ensures consistency in mathematical calculations. Without it, the same problem could have multiple answers, which would be super confusing! The order of operations, especially when dealing with multiplication, division, addition, and subtraction, is crucial. Following the PEMDAS rules ensures that we perform operations in the correct sequence, leading to the accurate and consistent results we need in simplifying algebraic expressions. This step-by-step approach not only simplifies the process but also clarifies the underlying mathematical principles. Think of this as the roadmap for your mathematical journey. Following the steps will lead you to the right destination every time, especially for more complex problems. Make sure to keep this order in mind when you are solving more complex equations, you'll be setting yourself up for success.

Conclusion: Your Simplified Expression!

Congratulations, guys! You have successfully simplified the expression 6p + 7q - 5q + 10. By combining the like terms, we found that the simplified form is 6p + 2q + 10. Remember, the key is to identify the like terms (terms with the same variables), combine their coefficients (add or subtract the numbers in front of the variables), and keep the variables and their exponents the same. The rest of the terms are constants, and you just add the constants with the variables. Keep practicing and applying these steps to simplify more complex expressions. Each problem that you solve will make you more confident. So, keep going. You're building a strong foundation for your algebra skills! You've successfully navigated the process of simplifying algebraic expressions. This skill is like a superpower in mathematics; it makes complex problems much easier to handle. You've also gained a solid understanding of like terms and the significance of the order of operations, laying a foundation for more advanced mathematical concepts. Keep practicing, and you'll become a pro in no time.