Simplifying Algebraic Expressions Step By Step Guide

Hey there, math enthusiasts! Ever find yourself staring at an algebraic expression that looks like it belongs in a sci-fi movie? Don't worry, we've all been there. The key to conquering these mathematical beasts is to break them down step by step, just like a detective solving a case. Today, we're going to dissect an expression that might seem daunting at first glance, but trust me, by the end of this article, you'll be simplifying expressions like a total pro. So, grab your pencils, put on your thinking caps, and let's dive into the world of algebraic simplification!

Decoding the Expression The Challenge Awaits

Let's face the challenge head-on. We're presented with the expression $3${\frac{7}{5} x+4}$-2${\frac{3}{2}-\frac{5}{4} x}$. This expression might seem like a jumbled mess of numbers, fractions, and variables, but fear not! Our mission, should we choose to accept it (and we do!), is to simplify this expression and find its true, simplified form. Before we jump into the nitty-gritty, let's take a moment to appreciate what we're dealing with. We have two sets of parentheses, each containing terms with the variable 'x' and constant terms. These parentheses are being multiplied by constants outside them. Our goal is to eliminate these parentheses, combine like terms, and present the expression in its simplest, most elegant form. Think of it as decluttering your mathematical space! Simplifying algebraic expressions is a fundamental skill in mathematics, paving the way for more advanced concepts like solving equations and inequalities. So, mastering this skill is crucial for your mathematical journey. And hey, it's not just about the math; it's about problem-solving, logical thinking, and attention to detail – skills that are valuable in all aspects of life.

The Distributive Property Our Secret Weapon

The distributive property is our secret weapon in this simplification quest. This property states that for any numbers a, b, and c, a(b + c) = ab + ac. In simpler terms, it means we can multiply the term outside the parentheses by each term inside the parentheses. This is the key to unlocking our expression and setting the terms free! Let's apply the distributive property to our expression. First, we'll distribute the 3 across the terms inside the first set of parentheses: 3 * (7/5 x) + 3 * 4. This gives us (21/5)x + 12. Next, we'll distribute the -2 across the terms inside the second set of parentheses: -2 * (3/2) - 2 * (-5/4 x). Notice the crucial negative sign in front of the 2 – we need to distribute that as well! This gives us -3 + (10/4)x, which simplifies to -3 + (5/2)x. Now, our expression looks a bit different. We've eliminated the parentheses, but we still have some work to do. We've got terms with 'x' and constant terms scattered around, and it's our job to bring them together, like organizing a messy room. The distributive property is a cornerstone of algebra, and understanding it is essential for simplifying expressions, solving equations, and tackling more complex mathematical problems. It's like having a superpower that allows you to break down barriers and conquer mathematical challenges. So, embrace the distributive property, practice it, and make it your ally in your mathematical adventures!

Combining Like Terms Bringing Order to Chaos

Now that we've unleashed the terms from their parentheses, it's time to combine like terms. Think of it as sorting your socks – you group the ones that are similar, right? We'll do the same with our algebraic terms. Like terms are terms that have the same variable raised to the same power. In our case, we have terms with 'x' and constant terms. Let's gather the 'x' terms: (21/5)x and (5/2)x. To combine them, we need a common denominator. The least common denominator for 5 and 2 is 10. So, we rewrite (21/5)x as (42/10)x and (5/2)x as (25/10)x. Now we can add them: (42/10)x + (25/10)x = (67/10)x. Great! We've tamed the 'x' terms. Next, let's gather the constant terms: 12 and -3. These are much easier to combine: 12 - 3 = 9. Fantastic! We've sorted the constants as well. Now, we have all the pieces of our simplified expression. We have the 'x' term, (67/10)x, and the constant term, 9. We simply combine them to get our final simplified expression. Combining like terms is a fundamental step in simplifying expressions, and it's crucial for solving equations and other algebraic problems. It's like putting the pieces of a puzzle together – you identify the pieces that belong together and connect them to form a complete picture. With practice, you'll become a master of combining like terms, and you'll be able to simplify expressions with confidence and ease.

The Grand Finale Revealing the Simplified Form

Drumroll, please! After our mathematical journey through distribution and combining like terms, we've arrived at the simplified form of our expression. Remember the original expression, $3${\frac{7}{5} x+4}$-2${\frac{3}{2}-\frac{5}{4} x}$ ? Well, after all our hard work, it transforms into the elegant expression $\frac{67}{10} x+9$. Isn't it satisfying to see a complex expression reduced to its simplest form? It's like taking a tangled mess of wires and neatly organizing them – everything is clear, concise, and easy to understand. But let's not just admire our result; let's make sure it's the correct one. We can double-check our work by plugging in a value for 'x' into both the original expression and the simplified expression. If we get the same result, we can be confident that our simplification is correct. This is a great way to verify your work and build confidence in your mathematical skills. The simplified expression, $\frac{67}{10} x+9$, is a linear expression in slope-intercept form, which is a standard and useful form in algebra. It allows us to easily identify the slope and y-intercept of the line represented by the expression. This is just one example of how simplifying expressions can lead to deeper understanding and insights in mathematics. So, celebrate your achievement! You've successfully simplified a complex algebraic expression, and you've gained valuable skills in the process. Remember, the journey of a thousand miles begins with a single step, and the journey of mathematical mastery begins with simplifying one expression at a time.

Choosing the Correct Answer Cracking the Code

Now that we've simplified the expression to $\frac{67}{10} x+9$, let's tackle the original question and choose the correct answer from the given options. We were presented with four options:

A. $-\frac{39}{5} x-\frac{11}{2}$

B. $\frac{67}{10} x+9$

C. $\frac{3}{10} x+\frac{5}{2}$

D.Discussion category : mathematics

By comparing our simplified expression with the options, it's clear that option B, $\frac{67}{10} x+9$, perfectly matches our result. Hooray! We've found the correct answer. But let's not just stop there. It's always a good idea to understand why the other options are incorrect. This can help us avoid making similar mistakes in the future. Option A has a negative coefficient for the 'x' term and a negative constant term, which doesn't match our positive coefficient and constant. Option C has a completely different coefficient for the 'x' term and a different constant term, indicating an error in the simplification process. Understanding why incorrect answers are wrong is just as important as knowing why the correct answer is right. It strengthens your understanding of the concepts and helps you develop critical thinking skills. So, always take the time to analyze the options and understand the reasoning behind the correct answer and the incorrect ones. This will make you a more confident and skilled problem-solver in mathematics and beyond.

Mastering Algebraic Simplification Your Path to Mathematical Prowess

Congratulations, mathletes! You've successfully navigated the world of algebraic simplification, conquering a complex expression and emerging victorious. You've learned the power of the distributive property, the art of combining like terms, and the importance of careful calculation. But remember, this is just one step on your mathematical journey. The more you practice, the more confident and skilled you'll become. So, keep simplifying expressions, keep solving equations, and keep exploring the fascinating world of mathematics. Think of algebraic simplification as a skill that unlocks doors to more advanced mathematical concepts. It's like building a strong foundation for a skyscraper – the stronger the foundation, the taller the skyscraper can be. By mastering algebraic simplification, you're building a solid foundation for your future mathematical endeavors. And remember, mathematics is not just about numbers and symbols; it's about problem-solving, logical thinking, and creative reasoning. These are skills that are valuable in all aspects of life, from everyday decision-making to complex scientific research. So, embrace the challenge, enjoy the journey, and never stop learning. The world of mathematics is vast and exciting, and with each step you take, you'll discover new wonders and unlock your full mathematical potential. So, go forth and conquer, and remember to always simplify with a smile!