Solving For X: -2.4x = 8.4 - Find The Value Of X

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In this comprehensive guide, we'll break down the steps to solve for x in the equation -2.4x = 8.4. Understanding how to solve linear equations is a fundamental skill in algebra, and this example provides a clear and practical application of algebraic principles. We will guide you through each step, ensuring that you not only arrive at the correct answer but also understand the underlying concepts. Let’s dive in and tackle this equation together!

Understanding the Basics of Linear Equations

Before we jump into solving the equation, let's quickly recap the basics of linear equations. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. The variable is raised to the power of one, and the equation, when graphed, forms a straight line. The goal in solving a linear equation is to isolate the variable on one side of the equation to determine its value.

In our case, the equation -2.4x = 8.4 fits this description perfectly. We have a variable x multiplied by a constant (-2.4), and our goal is to find the value of x that makes the equation true. To achieve this, we'll use the properties of equality, which allow us to perform the same operation on both sides of the equation without changing its balance. This is a cornerstone of algebraic manipulation, and mastering it is crucial for solving more complex problems.

Why is Solving Linear Equations Important?

Solving linear equations is not just an academic exercise; it has practical applications in various fields. From calculating finances and budgeting to understanding physics and engineering problems, the ability to manipulate and solve equations is invaluable. In real-world scenarios, you might use linear equations to determine costs, plan projects, or even optimize resources. The skills you gain from solving basic equations like this one form the foundation for more advanced mathematical concepts and problem-solving techniques.

Step-by-Step Solution for -2.4x = 8.4

Now, let's get to the heart of the matter and solve the equation -2.4x = 8.4 step by step. Our primary objective is to isolate x on one side of the equation. In this case, x is being multiplied by -2.4. To isolate x, we need to perform the inverse operation, which is division. We will divide both sides of the equation by -2.4.

  1. Write down the equation: -2. 4x = 8.4

  2. Divide both sides by -2.4: (-2.4x) / -2.4 = 8.4 / -2.4

This step is crucial because it maintains the balance of the equation. Whatever operation we perform on one side, we must perform on the other to keep the equation valid. By dividing both sides by -2.4, we are effectively undoing the multiplication that is affecting x. This is a fundamental principle in solving any algebraic equation.

  1. Simplify: x = -3.5

After performing the division, we find that x equals -3.5. This is our solution. We have successfully isolated x and determined its value. But our job isn't quite done yet. It's always a good idea to check our solution to ensure it's correct.

Verifying the Solution

To verify our solution, we substitute x = -3.5 back into the original equation -2.4x = 8.4 and see if it holds true.

  1. Substitute x = -3.5 into the original equation: -2. 4 * (-3.5) = 8.4

  2. Multiply -2.4 by -3.5: 8. 4 = 8.4

As we can see, the left side of the equation equals the right side, which confirms that our solution x = -3.5 is indeed correct. This step is a vital part of the problem-solving process. It not only gives us confidence in our answer but also helps us catch any potential errors we might have made along the way.

Common Mistakes to Avoid

Solving linear equations can sometimes be tricky, and it's easy to make mistakes if you're not careful. Here are a few common pitfalls to watch out for:

  • Forgetting the sign: When dividing or multiplying by a negative number, it’s essential to pay attention to the signs. A common mistake is to forget the negative sign, which can lead to an incorrect solution. Remember, a negative number divided by a negative number yields a positive result, and a positive number divided by a negative number yields a negative result.
  • Not performing the same operation on both sides: The golden rule of solving equations is to perform the same operation on both sides. Failing to do so will disrupt the balance of the equation and lead to a wrong answer. Whether it's addition, subtraction, multiplication, or division, make sure you apply it consistently to both sides.
  • Incorrect arithmetic: Simple arithmetic errors can derail your solution. Double-check your calculations, especially when dealing with decimals or fractions. Using a calculator can help reduce the risk of errors.
  • Skipping steps: While it might be tempting to skip steps to save time, doing so can increase the likelihood of making a mistake. Writing out each step clearly helps you keep track of your progress and identify any errors more easily. A methodical approach is often more efficient in the long run.

Practice Problems

To reinforce your understanding, let’s look at a few practice problems that are similar to the one we just solved. Working through these examples will help solidify your skills and build your confidence in solving linear equations.

  1. Solve for y: 3.2y = -9.6
  2. Solve for z: -1.5z = -6.0
  3. Solve for a: 4.8a = 14.4

For each of these problems, follow the same steps we used earlier: divide both sides of the equation by the coefficient of the variable to isolate the variable. Remember to pay attention to the signs and double-check your arithmetic. Practice makes perfect, and the more you work with these types of equations, the more comfortable you’ll become with the process.

Solutions to Practice Problems

Here are the solutions to the practice problems:

  1. y = -3
  2. z = 4
  3. a = 3

Take a moment to compare your solutions with these. If you arrived at the correct answers, congratulations! If not, go back and review your steps to identify where you might have made a mistake. Understanding where you went wrong is just as important as getting the right answer, as it helps you avoid similar errors in the future.

Real-World Applications

Solving linear equations isn't just a theoretical exercise; it has numerous practical applications in everyday life. Let’s explore some real-world scenarios where this skill can come in handy.

  • Finance: Imagine you’re saving up for a new gadget that costs $350, and you save $25 each week. You can use a linear equation to determine how many weeks it will take to save enough money. If w represents the number of weeks, the equation would be 25w = 350. Solving for w gives you the number of weeks you need to save.
  • Cooking: Suppose you’re doubling a recipe that calls for 1.5 cups of flour. To find out how much flour you need, you can use the equation 2 * 1.5 = x, where x is the amount of flour needed. Solving this simple equation tells you that you need 3 cups of flour.
  • Travel: If you’re driving at an average speed of 60 miles per hour and need to travel 300 miles, you can use a linear equation to calculate how long the trip will take. The equation would be 60t = 300, where t is the time in hours. Solving for t tells you that the trip will take 5 hours.

These examples illustrate how linear equations are embedded in our daily routines. Recognizing and solving these equations can help you make informed decisions and solve practical problems efficiently. From budgeting and cooking to planning trips and managing projects, the ability to work with linear equations is a valuable asset.

Conclusion: Mastering Linear Equations

In conclusion, solving for x in the equation -2.4x = 8.4 is a fundamental exercise in algebra that highlights the importance of understanding and applying algebraic principles. By following a step-by-step approach, paying attention to detail, and verifying our solution, we’ve not only found the value of x but also reinforced our problem-solving skills.

Remember, the key to mastering linear equations is practice. The more you work through problems, the more comfortable and confident you’ll become. Don't be afraid to make mistakes; they are valuable learning opportunities. Each problem you solve brings you one step closer to mastering this essential skill.

We hope this guide has been helpful in clarifying the process of solving linear equations. Keep practicing, keep exploring, and you’ll find that algebra becomes less daunting and more empowering. With a solid foundation in algebra, you’ll be well-equipped to tackle more complex mathematical challenges and real-world problems.

For further exploration and practice on algebraic equations, consider visiting Khan Academy's Algebra Section.