Simplifying Algebraic Expressions: A Step-by-Step Guide

Hey math enthusiasts! Today, we're diving into the world of algebraic expressions, specifically the simplification of the expression: $\frac{1}{3}(9x - 15) + 4(x + \frac{1}{2})$. We'll break down each step, making sure you understand the 'why' behind the 'how'. So, grab your pencils (or your favorite digital pen!), and let's get started. We'll be following Ranger's first two steps, and then expanding on them to solve the problem completely. This guide will walk you through the process, ensuring you understand each operation and how to apply them. It's all about making complex-looking problems manageable, one step at a time.

Step 1: Distributing the Terms – The Key to Unlocking the Expression

Alright, guys, the first step is all about distribution. What does that mean? It means taking the numbers outside the parentheses and multiplying them by each term inside the parentheses. So, for the expression $\frac{1}{3}(9x - 15) + 4(x + \frac{1}{2})$, we'll distribute $\frac{1}{3}$ across the terms inside the first set of parentheses $(9x - 15)$ and distribute $4$ across the terms in the second set of parentheses $(x + \frac{1}{2})$.

Let's break it down:

• For the first part, $\frac{1}{3}(9x - 15)$: We multiply $\frac{1}{3}$ by $9x$ and then by $-15$.
  • $\frac{1}{3} * 9x = 3x$ (because $\frac{9}{3} = 3$)
  • $\frac{1}{3} * -15 = -5$ (because $\frac{-15}{3} = -5$)
• For the second part, $4(x + \frac{1}{2})$: We multiply $4$ by $x$ and then by $\frac{1}{2}$.
  • $4 * x = 4x$
  • $4 * \frac{1}{2} = 2$ (because $4 / 2 = 2$)

Combining these results, the expression becomes: $3x - 5 + 4x + 2$. Notice that the parentheses are gone. That's the beauty of distribution; it allows us to remove them and work with individual terms. This step is absolutely crucial because it simplifies the expression, making it easier to combine like terms in the next phase. Think of it like taking apart a complicated machine to see all the individual parts before putting them back together (hopefully in a simpler way!). The main idea here is to eliminate the parentheses, preparing the expression for further simplification. Remembering the distributive property is vital; it is a fundamental tool for manipulating algebraic expressions. The goal is to prepare for the final stage of simplification, making it as easy as possible.

Step 2: Calculating Products – The Foundation of Simplification

In this step, we'll perform the multiplications identified in the first step. You've already done the hard work, calculating each product in the distribution process. The original expression $\frac{1}{3}(9x - 15) + 4(x + \frac{1}{2})$ transforms into the expanded form you see after distribution: $3x - 5 + 4x + 2$. Now, this step simply involves stating those products explicitly.

• rac{1}{3}(9x) becomes $3x$
• rac{1}{3}(-15) becomes $-5$
• $4(x)$ becomes $4x$
• $4(\frac{1}{2})$ becomes $2$

It is essential to understand that step 2 is really just a restatement of the products after distribution, making the transformation explicit. This helps in keeping track of all the operations. At this point, the expression has been expanded, the parentheses are gone, and we are one step closer to the final simplified form. This step, while seemingly small, is incredibly important for maintaining clarity and avoiding errors. The goal is always to reduce the expression to its simplest form, and this step lays the groundwork for that final reduction. By calculating products, we isolate the components of our expression, getting us closer to a solution. This is about ensuring each calculation is accurate and properly reflected in the new, more accessible expression. The focus remains on making sure the algebraic terms are correctly multiplied and that no components are overlooked.

Step 3: Combining Like Terms – Bringing It All Together

Now comes the fun part: combining like terms. Like terms are terms that have the same variable raised to the same power. In our simplified expression, $3x - 5 + 4x + 2$, the like terms are $3x$ and $4x$ (both have the variable $x$) and $-5$ and $2$ (both are constants - just numbers). Combining like terms simplifies the expression into a more manageable form. So let’s combine them.

• Combining the x terms: $3x + 4x = 7x$
• Combining the constant terms: $-5 + 2 = -3$

By putting these together, our simplified expression becomes $7x - 3$. This is our final, simplified answer! We've taken a more complex algebraic expression and transformed it into a much simpler, more understandable one. This step streamlines the expression, making it easier to interpret and use in further calculations. It’s a fundamental technique used across algebra. Remember, the goal is always to make expressions as simple as possible without changing their value.

Step 4: The Final Simplified Expression – The Answer Revealed

After combining like terms, we arrive at our final, simplified expression: $7x - 3$. This is the equivalent of the original expression, but in a much more concise form. It's the answer we've been working toward. This simplified form is easier to work with. If we were to substitute a value for x, it would be much simpler to calculate the result using $7x - 3$ than using the original, more complex expression. The transformation from the original expression, through the distributive property and combining like terms, to $7x - 3$ represents a complete simplification. This final step confirms the accuracy of all preceding steps. The ability to manipulate algebraic expressions is a core skill in mathematics, enabling you to solve problems efficiently and understand the underlying relationships between variables and numbers. The simplification is complete, and we have a single, clean expression representing the original.

Why This Matters – The Big Picture

Simplifying algebraic expressions is a fundamental skill in mathematics. It's not just about solving this one problem; it's about building a foundation for more advanced mathematical concepts. This skill is critical for:

• Solving equations: Simplification is essential for isolating variables and finding solutions.
• Working with formulas: Simplifying formulas makes them easier to understand and apply.
• Calculus and beyond: A strong grasp of simplification techniques is necessary for success in more advanced math courses.

So, whether you're a student, a professional, or just someone who enjoys a mental challenge, mastering algebraic simplification will serve you well. It's a key skill that builds confidence and opens doors to more complex mathematical explorations. The principles we have covered extend far beyond the immediate problem. They serve as essential tools in various mathematical applications. Keep practicing; the more you simplify, the more comfortable and proficient you'll become!

I hope this step-by-step guide has been helpful! If you have any questions or want to try another example, just let me know. Happy simplifying, everyone!