Unlocking The Equation: A Step-by-Step Guide

by Alex Johnson 45 views

Welcome, math enthusiasts! Today, we're diving into the world of algebraic equations. Specifically, we'll tackle the equation: x3−x−56=5x2+1\frac{x}{3}-\frac{x-5}{6}=\frac{5 x}{2}+1. Don't worry if it looks a bit intimidating at first glance; we'll break it down step-by-step, making it easy to understand and solve. Equations like these are fundamental in algebra and are essential for various applications in science, engineering, and everyday problem-solving. Mastering the art of solving equations is like having a superpower – it allows you to decipher complex problems and find the hidden solutions.

Understanding the Basics: Equations and Variables

Before we jump into the equation, let's refresh our understanding of some core concepts. An equation is a mathematical statement that asserts the equality of two expressions. It always contains an equals sign (=), indicating that the values on either side are the same. In our equation, the expression on the left-hand side is x3−x−56\frac{x}{3}-\frac{x-5}{6}, and the expression on the right-hand side is 5x2+1\frac{5 x}{2}+1. Our goal is to find the value(s) of the variable, in this case, 'x', that make the equation true.

The variable 'x' represents an unknown quantity. It's the puzzle piece we need to find to solve the equation. In this specific equation, 'x' appears in several terms, making it a bit more complex than a simple one-step equation. However, by applying a series of logical steps and algebraic rules, we can isolate 'x' and determine its value. The key is to maintain the balance of the equation throughout the process. Whatever operation we perform on one side, we must also perform on the other side to ensure the equation remains valid. Think of it like a seesaw – to keep it balanced, you must add or remove weight from both sides equally. Understanding the role of the variable and the concept of balance is fundamental to solving any algebraic equation. This foundational knowledge will pave the way for tackling more complex equations in the future. Mastering these basics is like building a strong foundation for a house – the stronger the foundation, the sturdier the structure.

Step-by-Step Solution: Unraveling the Equation

Now, let's get down to the exciting part: solving the equation! We'll follow a series of steps to isolate 'x' and find its value. Remember, the goal is to get 'x' by itself on one side of the equation. Here's the breakdown:

  1. Eliminate the Fractions: The first step is to get rid of those pesky fractions. To do this, we need to find the least common multiple (LCM) of the denominators (3, 6, and 2). The LCM of 3, 6, and 2 is 6. We'll multiply every term in the equation by 6. This is allowed because multiplying both sides of an equation by the same number doesn't change the equality.

    So, the equation x3−x−56=5x2+1\frac{x}{3}-\frac{x-5}{6}=\frac{5 x}{2}+1 becomes:

    6⋅x3−6⋅x−56=6⋅5x2+6⋅16 \cdot \frac{x}{3} - 6 \cdot \frac{x-5}{6} = 6 \cdot \frac{5x}{2} + 6 \cdot 1

    This simplifies to:

    2x−(x−5)=15x+62x - (x-5) = 15x + 6

    Notice that the fractions are gone! This is a significant simplification.

  2. Simplify and Distribute: Next, we'll simplify the equation by distributing the negative sign in the term −(x−5)-(x-5).

    The equation 2x−(x−5)=15x+62x - (x-5) = 15x + 6 becomes:

    2x−x+5=15x+62x - x + 5 = 15x + 6

    Combine like terms on the left side:

    x+5=15x+6x + 5 = 15x + 6

  3. Isolate the Variable (x): Our next goal is to get all the 'x' terms on one side of the equation and the constant terms (numbers without 'x') on the other side. Let's subtract 'x' from both sides:

    x+5−x=15x+6−xx + 5 - x = 15x + 6 - x

    This simplifies to:

    5=14x+65 = 14x + 6

    Now, subtract 6 from both sides:

    5−6=14x+6−65 - 6 = 14x + 6 - 6

    Which gives us:

    −1=14x-1 = 14x

  4. Solve for x: Finally, we isolate 'x' by dividing both sides of the equation by 14:

    −114=14x14\frac{-1}{14} = \frac{14x}{14}

    Therefore:

    x=−114x = -\frac{1}{14}

    And there we have it! The solution to the equation is x=−114x = -\frac{1}{14}. Understanding each step is crucial to solving the equation correctly.

Verification: Ensuring the Solution is Correct

It's always a good practice to verify your solution to ensure its accuracy. We can do this by substituting the value of 'x' we found back into the original equation and checking if both sides are equal. Let's substitute x=−114x = -\frac{1}{14} into the original equation:

x3−x−56=5x2+1\frac{x}{3}-\frac{x-5}{6}=\frac{5 x}{2}+1

−1143−−114−56=5⋅−1142+1\frac{-\frac{1}{14}}{3}-\frac{-\frac{1}{14}-5}{6}=\frac{5 \cdot -\frac{1}{14}}{2}+1

Simplifying each term:

−142−−71146=−528+1-\frac{1}{42} - \frac{-\frac{71}{14}}{6} = -\frac{5}{28} + 1

−142+7184=−528+1-\frac{1}{42} + \frac{71}{84} = -\frac{5}{28} + 1

Finding a common denominator and simplifying further:

−284+7184=−528+2828\frac{-2}{84} + \frac{71}{84} = \frac{-5}{28} + \frac{28}{28}

6984=2328\frac{69}{84} = \frac{23}{28}

2328=2328\frac{23}{28} = \frac{23}{28}

Since both sides of the equation are equal, we can confidently say that our solution, x=−114x = -\frac{1}{14}, is correct. This verification step is crucial as it helps build confidence in your mathematical abilities and ensures that any errors made during the solving process are caught early on. Always remember to double-check your work!

Tips and Tricks for Solving Equations

Solving equations can become much easier with a few handy tips and tricks. Firstly, practice is key. The more equations you solve, the more comfortable and proficient you'll become. Start with simpler equations and gradually work your way up to more complex ones. Secondly, always double-check your work. This helps catch any calculation errors and ensures you arrive at the correct solution. Thirdly, organize your steps. Write down each step clearly and neatly, which makes it easier to follow your work and identify any mistakes. Use a pencil and paper to keep track of your calculations. Fourthly, understand the properties of equality. Knowing the properties allows you to manipulate equations without changing their value. Finally, don't be afraid to ask for help. If you're struggling with a particular problem, don't hesitate to seek assistance from a teacher, tutor, or online resources. There are plenty of resources available to help you. These are the essential tools for mastering equations.

Conclusion: Your Journey in Algebra

Congratulations! You've successfully solved a linear equation with fractions. This is a significant accomplishment in your algebra journey. Remember that solving equations is a fundamental skill in mathematics, and with practice, you can master it. Keep practicing, stay curious, and don't be afraid to challenge yourself with more complex problems. Every equation you solve is a step forward in your mathematical journey. Algebra opens doors to understanding the world around us. From modeling real-world problems to predicting future trends, the power of algebra is immense. So, embrace the challenge, enjoy the process, and keep exploring the fascinating world of mathematics. Continue to hone your skills as you progress. Your mathematical journey has just begun!

To further enhance your understanding and explore more related topics, consider visiting these trusted resources:

  • Khan Academy: https://www.khanacademy.org/ - Khan Academy provides comprehensive lessons and practice exercises on algebra and other mathematical topics.