Solve 4x * 26 = 8 - 5x: Step-by-Step Guide

Hey everyone! Today, we're diving into a fun math problem: solving the equation 4x * 26 = 8 - 5x. Don't worry, it's not as intimidating as it looks! We'll break it down step by step, so you can easily understand how to find the value of 'x'. We'll explore each step in detail, ensuring you grasp the underlying concepts and can confidently tackle similar problems in the future. Math can be enjoyable, especially when you understand the process. So, let's get started and unravel this equation together!

Understanding the Equation

Before we jump into solving, let's first understand what the equation 4x * 26 = 8 - 5x is telling us. In simple terms, an equation is a mathematical statement that shows two expressions are equal. Our goal here is to find the value of 'x' that makes this statement true. This involves manipulating the equation while maintaining the equality to isolate 'x' on one side. Equations are the foundation of algebra and are used extensively in various fields, from science and engineering to economics and computer science. Mastering the art of solving equations opens doors to a deeper understanding of these disciplines. So, let's embark on this journey of mathematical discovery and see how we can unlock the value of 'x' in our equation. Remember, every equation is a puzzle waiting to be solved, and with the right tools and approach, we can find the solution.

Step 1: Simplify the Equation

The first thing we need to do is simplify both sides of the equation. On the left side, we have 4x * 26, which can be multiplied to give us 104x. So, our equation now looks like this: 104x = 8 - 5x. Simplifying equations is like tidying up a room; it makes everything clearer and easier to work with. In this case, multiplying the terms on the left side helps us consolidate the expression and move closer to isolating 'x'. This step highlights the importance of basic arithmetic operations in algebra. By performing these operations correctly, we pave the way for the subsequent steps in solving the equation. Think of it as building a strong foundation for the rest of the solution. Without this simplification, the equation might appear more complex than it actually is. So, always look for opportunities to simplify expressions before proceeding further. It's a fundamental technique that will serve you well in your mathematical journey.

Step 2: Isolate the 'x' Terms

Now, we want to get all the terms with 'x' on one side of the equation. To do this, we can add 5x to both sides. This gives us: 104x + 5x = 8 - 5x + 5x, which simplifies to 109x = 8. The idea behind isolating variables is to create a situation where we have the variable we're trying to solve for on its own. By adding 5x to both sides, we effectively cancel out the -5x on the right side, bringing all the 'x' terms to the left. This is a crucial step in solving any algebraic equation. It's like separating ingredients in a recipe so you can combine them in the right proportions. By isolating 'x', we're setting the stage for the final step, where we'll determine its value. This technique demonstrates the power of inverse operations in algebra. We use addition to undo subtraction and vice versa. Mastering this concept is key to becoming proficient in solving equations.

Step 3: Solve for 'x'

We're almost there! We have 109x = 8. To find the value of 'x', we need to divide both sides of the equation by 109. This gives us: x = 8 / 109. So, the solution is x = 8/109. Dividing both sides by the coefficient of 'x' is the final step in isolating the variable and finding its value. This operation undoes the multiplication that was previously applied to 'x'. It's like peeling away the layers to reveal the core value. In this case, we find that 'x' is equal to the fraction 8/109. While this might not be a whole number, it's a perfectly valid solution. It's important to remember that solutions to equations can be fractions, decimals, or even irrational numbers. The key is to follow the steps correctly and apply the principles of algebra. This final step highlights the importance of understanding inverse operations and their role in solving equations. By mastering this technique, you'll be well-equipped to tackle a wide range of algebraic problems.

Checking the Answer

It's always a good idea to check our answer to make sure it's correct. We can do this by substituting x = 8/109 back into the original equation: 4x * 26 = 8 - 5x. So, 4 * (8/109) * 26 should equal 8 - 5 * (8/109). Let's calculate:

• Left side: 4 * (8/109) * 26 = (4 * 8 * 26) / 109 = 832 / 109
• Right side: 8 - 5 * (8/109) = 8 - (40/109) = (8 * 109 - 40) / 109 = (872 - 40) / 109 = 832 / 109

Both sides are equal! This confirms that our solution x = 8/109 is correct. Checking your answer is like proofreading a document before submitting it; it helps catch any errors and ensures accuracy. By substituting the solution back into the original equation, we verify that it satisfies the equality. This step reinforces the understanding of what it means to solve an equation – to find a value that makes the equation true. It also provides an opportunity to review the calculations and ensure that no mistakes were made along the way. This practice builds confidence in your problem-solving skills and helps develop a habit of thoroughness in mathematics. So, always take the time to check your answers; it's a valuable investment in your mathematical journey.

Why is it Important to Solve Equations?

Solving equations is a fundamental skill in mathematics and has countless applications in real life. From calculating the trajectory of a rocket to determining the optimal dosage of medication, equations are used to model and solve problems in various fields. Understanding how to solve equations not only enhances your mathematical abilities but also equips you with valuable problem-solving skills that are transferable to other areas of life. Equations are the language of science and engineering, allowing us to describe relationships between quantities and make predictions. They are also used in economics, finance, and computer science to model complex systems and make informed decisions. So, mastering the art of solving equations is an investment in your future, opening doors to a wide range of opportunities and empowering you to tackle challenges with confidence. It's like learning a new language that allows you to communicate with the world in a more precise and meaningful way. So, embrace the power of equations and unlock their potential to solve problems and make a difference.

Common Mistakes to Avoid

When solving equations, there are a few common mistakes that students often make. One mistake is not applying operations to both sides of the equation. Remember, to maintain equality, whatever you do to one side, you must do to the other. Another common mistake is making errors in arithmetic, such as adding or subtracting numbers incorrectly. Always double-check your calculations to avoid these mistakes. Additionally, students sometimes forget to distribute properly when dealing with parentheses or fail to combine like terms correctly. These errors can lead to incorrect solutions. To avoid these pitfalls, it's crucial to practice solving equations regularly and to pay close attention to detail. Taking your time and working through each step carefully can significantly reduce the chances of making mistakes. Also, don't hesitate to ask for help if you're struggling with a particular concept or technique. Learning from your mistakes is an essential part of the mathematical journey, so embrace the challenges and strive for accuracy and understanding.

Conclusion

So, we've successfully solved the equation 4x * 26 = 8 - 5x and found that x = 8/109. Remember, the key to solving equations is to simplify, isolate the variable, and check your answer. With practice, you'll become a pro at solving all kinds of equations! Keep practicing, and you'll find that solving equations becomes second nature. The more you work with them, the more comfortable and confident you'll become. And remember, math is not just about finding the right answer; it's about the process of problem-solving and the logical thinking it develops. So, embrace the challenges, celebrate your successes, and enjoy the journey of mathematical discovery. You've got this!