Solving The Equation: A Step-by-Step Guide

by Alex Johnson 43 views

Introduction: Unraveling the Equation

Hey there, math enthusiasts! Today, we're diving into a classic algebra problem: solving the equation x4+x5=6320{\frac{x}{4} + \frac{x}{5} = \frac{63}{20}}. Don't worry if it looks a bit intimidating at first glance; we'll break it down into easy-to-understand steps. This type of equation, involving fractions and a single variable (in this case, 'x'), is fundamental in algebra. Understanding how to solve it is crucial for building a strong foundation in mathematics. We'll explore the core concepts and techniques needed to find the value of 'x' that makes this equation true. Think of it as a mathematical puzzle – and we're about to solve it!

Solving equations like this one is a fundamental skill in mathematics. It's used everywhere, from basic arithmetic to advanced calculus, and in fields that include physics, engineering, and computer science. The main goal here is to isolate the variable 'x' on one side of the equation. This involves manipulating the equation using algebraic rules while ensuring that both sides of the equation remain equal. The techniques we will use are simplification, multiplication, and division. Understanding how these tools work is key to solving the equation efficiently and accurately. So, let’s get started. We'll first address the fractions involved to simplify the process. This involves finding a common denominator and combining the fractions, and we will get the final answer after this series of steps. By the end of this guide, you will be able to solve similar equations with confidence!

We start with the equation x4+x5=6320{\frac{x}{4} + \frac{x}{5} = \frac{63}{20}}. The first thing that we must do is to find a common denominator for the fractions on the left side of the equation. The denominators are 4 and 5. The least common multiple (LCM) of 4 and 5 is 20, because 4 multiplied by 5 is 20. And 5 multiplied by 4 is 20. Now we can rewrite the equation. We convert x4{\frac{x}{4}} to an equivalent fraction with a denominator of 20 by multiplying both the numerator and denominator by 5, which gives us 5x20{\frac{5x}{20}}. We convert x5{\frac{x}{5}} to an equivalent fraction with a denominator of 20 by multiplying both the numerator and denominator by 4, which gives us 4x20{\frac{4x}{20}}. Therefore, the equation becomes 5x20+4x20=6320{\frac{5x}{20} + \frac{4x}{20} = \frac{63}{20}}. The next step is to combine the fractions on the left side, so add the numerators. So, 5x + 4x = 9x. The equation then becomes 9x20=6320{\frac{9x}{20} = \frac{63}{20}}. Now the fractions have the same denominator, which makes it easier to solve for x.

Step-by-Step Solution

Let's get down to the nitty-gritty and work through the steps to solve the equation. We will be using the concepts of fractions, and isolating the variable. Remember, the goal is always to get 'x' by itself on one side of the equation. Here’s a detailed breakdown:

  1. Find the Common Denominator: The original equation is x4+x5=6320{\frac{x}{4} + \frac{x}{5} = \frac{63}{20}}. As mentioned before, the common denominator for the fractions on the left side is 20. We rewrite the fractions with this common denominator: 5x20+4x20=6320{\frac{5x}{20} + \frac{4x}{20} = \frac{63}{20}}.

  2. Combine Like Terms: With the common denominator, we can now combine the fractions on the left side of the equation. This involves adding the numerators: 5x+4x=9x{5x + 4x = 9x}. The equation simplifies to 9x20=6320{\frac{9x}{20} = \frac{63}{20}}.

  3. Eliminate the Denominator: To eliminate the denominator, we can multiply both sides of the equation by 20. This gives us:

    • 20⋅9x20=20⋅6320{20 \cdot \frac{9x}{20} = 20 \cdot \frac{63}{20}}
    • Which simplifies to 9x=63{9x = 63}.

  4. Isolate 'x': To isolate 'x', we need to get rid of the coefficient 9. We do this by dividing both sides of the equation by 9:

    • 9x9=639{\frac{9x}{9} = \frac{63}{9}}
    • This gives us x=7{x = 7}.

Therefore, the solution to the equation x4+x5=6320{\frac{x}{4} + \frac{x}{5} = \frac{63}{20}} is x=7{x = 7}. We now know that the value of 'x' that satisfies the original equation is 7. We can check our work to ensure that we are right by substituting 7 in for 'x' in the original equation and verify that both sides are equal.

Let's check our work by substituting x=7{x = 7} into the original equation:
74+75=6320{\frac{7}{4} + \frac{7}{5} = \frac{63}{20}}

First, we calculate 74{\frac{7}{4}}, which equals 1.75. Second, we calculate 75{\frac{7}{5}}, which equals 1.4. Then, add 1.75 and 1.4 together, which equals 3.15. Finally, we must calculate 6320{\frac{63}{20}}, which also equals 3.15. Since both sides of the equation equal 3.15, we can confirm our solution of x = 7 is correct. This is how you verify the solution is accurate.

Key Concepts in Solving Equations

Before we dive deeper, let’s make sure we have a clear understanding of the key concepts involved in solving these kinds of equations. We need to know how to find a common denominator, combine like terms, and isolate the variable.

  • Common Denominator: When adding or subtracting fractions, the common denominator is the foundation. It's the least common multiple (LCM) of the denominators of the fractions. Finding a common denominator allows you to combine fractions easily, making the rest of the solving process possible. In the example above, the common denominator was 20. Finding the common denominator is a critical skill for simplifying expressions and equations containing fractions.
  • Combining Like Terms: Like terms are terms that have the same variable raised to the same power. They can be added or subtracted to simplify the equation. In our equation, after converting the fractions to have the same denominator, we were able to combine the terms 5x{5x} and 4x{4x} because they both involved the variable 'x' to the first power. Combining like terms streamlines the equation and makes it easier to solve.
  • Isolating the Variable: This is the ultimate goal. To isolate the variable means to get it alone on one side of the equation. We achieve this using inverse operations (doing the opposite operation) on both sides of the equation. If a term is added, we subtract it from both sides; if a term is multiplied, we divide both sides. Isolating the variable enables us to find its numerical value, which is the solution to the equation. Keep in mind that whatever operation is performed on one side of the equation must be performed on the other side to maintain equality.

By keeping these concepts in mind, you will be able to solve these types of equations with more confidence. Remember to always double-check your work to ensure accuracy!

Tips and Tricks for Solving Equations

Here are some helpful tips and tricks to make solving equations easier and to avoid common mistakes. These tips will help streamline the process and boost your accuracy.

  1. Practice Regularly: The more you solve equations, the more familiar you will become with the steps and the different types of equations. Practice makes perfect!
  2. Double-Check Your Work: Always verify your solution by substituting the value of 'x' back into the original equation. This is a simple but effective way to catch errors. Take your time when checking your work. Rushing will cause you to miss errors.
  3. Simplify First: Before starting to solve, simplify both sides of the equation as much as possible. Combine like terms and perform any necessary arithmetic operations. This will help reduce the chance of making mistakes.
  4. Keep Track of Signs: Be very careful with positive and negative signs. A small mistake in signs can lead to a wrong answer. Double-check your signs at every step.
  5. Use a Systematic Approach: Always follow a consistent set of steps. Writing down each step can help keep you organized and reduce the chances of missing something. Don't try to take shortcuts, especially when you are just beginning to learn.
  6. Break Down Complex Problems: If the equation seems too complex, break it down into smaller, more manageable parts. Solve each part separately and then combine the results.
  7. Know Your Basic Math Facts: Make sure you are strong in your multiplication tables, and other basic arithmetic facts. These fundamentals are essential for efficient problem-solving.

These tips are designed to make solving equations a smoother and less intimidating experience. Incorporating these strategies will improve your problem-solving skills.

Common Mistakes to Avoid

Even seasoned math students can make mistakes. Recognizing common pitfalls can prevent errors and improve your problem-solving accuracy. Let's look at some common mistakes:

  1. Incorrectly Finding the Common Denominator: This is a frequent error. Make sure to find the least common multiple (LCM) of the denominators. Using a larger common denominator than necessary complicates calculations and increases the risk of mistakes.
  2. Forgetting to Multiply All Terms: When you multiply both sides of an equation by a number, be sure to multiply every term. A common error is only multiplying one part of the equation.
  3. Mistakes with Signs: Positive and negative signs can trip up even the most careful students. Double-check your work when adding, subtracting, multiplying, and dividing positive and negative numbers.
  4. Incorrectly Combining Like Terms: Only combine terms that have the same variable raised to the same power. For example, you cannot combine x{x} and x2{x^2}. Ensure that you only combine like terms correctly.
  5. Not Checking Your Solution: After finding a solution, many students fail to check it. Always substitute the value of the variable back into the original equation to verify that it is correct. This is the best way to prevent errors.
  6. Making Arithmetic Errors: Simple arithmetic errors, such as miscalculating basic addition or multiplication, can lead to the wrong answer. Take your time and double-check each calculation.
  7. Skipping Steps: While it's tempting to take shortcuts, skipping steps can often lead to mistakes. Write out each step, particularly when solving more complex equations, to stay organized and reduce errors. Following these guidelines will help minimize errors and maximize your accuracy.

Conclusion: Mastering the Equation

Congratulations! You've successfully navigated the process of solving the equation x4+x5=6320{\frac{x}{4} + \frac{x}{5} = \frac{63}{20}}. We started with a seemingly complex equation and broke it down into manageable steps, highlighting the importance of finding a common denominator, combining like terms, and isolating the variable. This journey isn't just about solving one particular problem; it’s about understanding the core concepts of algebra and building a skill set that can be applied to many other mathematical challenges.

Remember, practice is key. The more you work through equations, the more confident you'll become. Each equation you solve is a step forward in strengthening your mathematical foundation. Embrace the challenges, learn from your mistakes, and continue to refine your skills. Keep these tips and strategies in mind as you tackle future equations, and you'll find yourself approaching problems with greater ease and precision. Math can be fun! Happy solving!

For more detailed information and additional examples, you can check out this resource: Khan Academy.