Simplifying Expressions: A Step-by-Step Guide

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Hey there, math enthusiasts! Ever wondered how to break down complex expressions into simpler forms? Well, you're in luck! Today, we're diving deep into the art of simplifying expressions. We'll be using the expression 9 + 5(4 + x) as our guinea pig, and together, we'll walk through each step, understanding the "why" behind every move. This isn't just about getting the right answer; it's about grasping the core principles that make algebra tick. So, grab your pencils, and let's get started. We'll be exploring the fundamental properties that govern mathematical operations, ensuring that the process is not just a series of actions but a journey of understanding. The goal is not merely to arrive at a solution but to equip you with the knowledge to approach any algebraic problem with confidence and clarity. The ability to simplify expressions is a cornerstone skill in mathematics, opening doors to more advanced concepts. The objective here is to make the process accessible and enjoyable, making math less daunting and more engaging. We are going to break down the process into easy-to-follow steps, with detailed explanations for each one, so you'll be an expert in no time. This method involves applying mathematical rules in a logical sequence to convert a complex expression into a simpler, more manageable form. Mastering this process is key to progressing in algebra and other mathematical disciplines. Let's start this adventure in the realm of numbers and variables.

The Given Expression and the First Transformation

Alright, let's begin with our starting point: the expression 9 + 5(4 + x). This is where it all starts. In the world of algebra, expressions can look a bit like puzzles, but fear not, we'll solve this one together. Initially, we have a combination of numbers, variables, and operations, and the goal is to make it simpler. The key here is to follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Always remember PEMDAS when solving, as it is the most important part of simplifying expressions. Now, the first step in simplifying involves dealing with the parentheses. Before we can do anything with the addition and the 9 outside, we must first address what's inside the parentheses. This is a fundamental concept in mathematical problem-solving; the parentheses dictate the sequence of operations. This directs us to focus on simplifying the terms inside before considering other operations. In our expression, (4 + x) sits inside the parentheses. Because we cannot simplify this further at this stage, we have to find another approach. That's where the distributive property comes into play. The distributive property allows us to multiply a term by everything inside the parentheses. So we'll have to use the distributive property to simplify this. The aim is to remove the parentheses, making the expression easier to work with. Remember, the goal is always to move towards a simpler form while maintaining mathematical accuracy. This foundational step sets the stage for the rest of the simplification process, making each subsequent step clearer and more manageable. The starting point provides the complexity, and with each step, we simplify towards the goal. The distributive property will be our primary tool at this stage.

Step 1: Applying the Distributive Property

Here, we use the distributive property to deal with the 5(4 + x) part of our expression. The distributive property says that a(b + c) = ab + ac. Applying this to our expression, we multiply the 5 by each term inside the parentheses. This means we multiply 5 by 4 and also 5 by x. The process transforms 5(4 + x) into (5 * 4) + (5 * x), or 20 + 5x. This is a critical step because it eliminates the parentheses, which is always the goal to simplify in the initial stages. By doing so, we are rearranging the expression in a way that allows us to combine like terms and simplify further. The distributive property is a fundamental rule, and understanding it is critical for algebra and beyond. This is why we focus on explaining it with as many details as possible, so that it becomes natural for you. Now, our expression becomes 9 + 20 + 5x. We have successfully applied the distributive property! This step is a direct application of the distributive property, ensuring that all terms within the parentheses are properly accounted for, and there are no hidden steps. Remember, mathematical accuracy is the name of the game!

Combining Like Terms and Final Simplification

Following the distributive property, the expression now looks like this: 9 + 20 + 5x. Great, now we move on to the next phase: combining like terms. This step is about identifying terms that can be added or subtracted because they share the same variable or are constants. Constants are the numbers without variables, while the other numbers have variables and also known as coefficients. In our expression, we have the constants 9 and 20. The term 5x stands alone because it has a variable, x, making it different from the constants. In order to simplify, we need to add the constants 9 and 20 together. This is a simple addition operation. Now, adding 9 and 20 gives us 29. The variable term, 5x, remains unchanged as there are no other x terms to combine it with. The strategy here is to keep simplifying by reducing the number of terms and making the expression as concise as possible. Simplifying the expression means rewriting it in its most compact form. That is what we are currently doing. The goal is always to make the final expression as clear and straightforward as possible. This step prepares the expression for its final form.

Step 2: Adding the Constants

Now, let's focus on combining the constant terms. We have 9 + 20 + 5x. To combine the constants, we add 9 and 20. Doing so, 9 + 20 equals 29. So, the expression now simplifies to 29 + 5x. At this point, no other simplification is possible. We cannot combine 29 and 5x because they are not like terms. The term 29 is a constant, while 5x is a term with a variable. Because they have no variables in common, they cannot be added directly. This step is all about making the expression more compact, resulting in 29 + 5x. The expression is now in its simplest form. We have done it! The process is now complete. You've reached the final form of the expression. Always remember this final form!

The Final Simplified Form

So, after all the steps, our original expression 9 + 5(4 + x) is now simplified to 29 + 5x. This result is a concise and manageable form of the original expression. The simplification process demonstrates how we can manipulate mathematical expressions, following specific rules to transform them. The transformation process highlights the power of algebra. The final form reveals a clear relationship between constants and the variable x. This final step underscores the efficiency and logic of mathematical simplification. This process also shows the power of mathematics. It takes an initially complex expression and converts it into a simpler one.

Conclusion

And there you have it, guys! We've successfully simplified the expression 9 + 5(4 + x) step by step, from the given form to its final, simplified form, which is 29 + 5x. It's a fantastic achievement! We did it by following some simple rules and understanding the properties of algebra. Remember, the distributive property and combining like terms are your friends. Keep practicing, and you'll become a simplification pro in no time! So, keep exploring the world of algebra, and have fun! You're now equipped with the knowledge and the skills to take on many more mathematical challenges. Just remember the steps and the rules, and you'll do great. Always remember PEMDAS and the Distributive Property.