Solving The Equation 5(x-4) = 3x + 4 A Step-by-Step Guide

Introduction

In this article, we will delve into the process of solving the equation 5(x - 4) = 3x + 4. This is a fundamental algebraic equation, and understanding how to solve it is crucial for mastering more complex mathematical concepts. We will break down the steps involved, providing a clear and concise explanation for each. This comprehensive guide aims to help students, educators, and anyone interested in mathematics to grasp the techniques required to solve such equations confidently. We'll cover everything from the initial distribution to the final verification of the solution. Through this step-by-step approach, we will not only find the correct answer but also reinforce the underlying principles of algebraic manipulation. This introduction sets the stage for a detailed exploration of the problem, ensuring that readers understand the context and importance of solving equations like this one.

Step 1: Distribute the 5

The first step in solving the equation 5(x - 4) = 3x + 4 is to eliminate the parentheses. We achieve this by distributing the 5 across the terms inside the parenthesis. This means we multiply both x and -4 by 5. When we multiply 5 by x, we get 5x. When we multiply 5 by -4, we get -20. Therefore, the left side of the equation becomes 5x - 20. Now, the equation looks like this: 5x - 20 = 3x + 4. This step is crucial because it simplifies the equation, making it easier to isolate the variable x. Distribution is a fundamental property in algebra, and mastering it is essential for solving various types of equations. By applying this step correctly, we set the stage for further simplification and ultimately finding the value of x. This initial distribution is the foundation upon which we build the rest of our solution, so it’s important to understand the mechanics and logic behind it. This distributive property allows us to rewrite the equation in a form that is more manageable and closer to our goal of isolating the variable.

Step 2: Rearrange the Equation

After distributing the 5, our equation is now 5x - 20 = 3x + 4. The next step involves rearranging the equation to get all the terms with x on one side and the constants on the other side. To do this, we need to subtract 3x from both sides of the equation. This maintains the equality and moves the 3x term from the right side to the left side. When we subtract 3x from 5x, we get 2x. So, the equation becomes 2x - 20 = 4. Now, we need to move the constant term -20 to the right side of the equation. To do this, we add 20 to both sides. This again maintains the equality and isolates the x term further. Adding 20 to 4 gives us 24. The equation is now simplified to 2x = 24. This rearrangement is a critical step in solving equations, as it groups like terms together, making it easier to isolate the variable. The key principle here is maintaining balance by performing the same operation on both sides of the equation. This rearrangement process is a cornerstone of algebraic manipulation and is essential for solving a wide range of equations.

Step 3: Isolate x

Now that we have the equation 2x = 24, our goal is to isolate x completely. To do this, we need to divide both sides of the equation by the coefficient of x, which is 2. When we divide 2x by 2, we are left with just x. When we divide 24 by 2, we get 12. Therefore, the solution to the equation is x = 12. This step is the final piece of the puzzle in solving for x. By dividing both sides by the coefficient of x, we effectively undo the multiplication and reveal the value of x. This process of isolating the variable is a fundamental technique in algebra and is crucial for solving equations of all types. Understanding this step allows us to confidently determine the value of the unknown in a given equation. This isolation process is the culmination of all our previous steps and provides the final answer to our problem.

Step 4: Verify the Solution

To ensure that our solution x = 12 is correct, we need to verify it by substituting this value back into the original equation: 5(x - 4) = 3x + 4. Substituting x = 12, we get 5(12 - 4) = 3(12) + 4. First, we simplify the expressions inside the parentheses: 12 - 4 = 8. So, the left side of the equation becomes 5(8), which equals 40. Next, we simplify the right side of the equation: 3(12) = 36, and then we add 4, so 36 + 4 = 40. Thus, the right side of the equation also equals 40. Since both sides of the equation are equal (40 = 40), our solution x = 12 is correct. This verification step is crucial in the problem-solving process. It provides a check on our work and ensures that the value we found for x truly satisfies the original equation. By substituting the solution back into the equation, we can confirm its validity and gain confidence in our answer. This verification process is a best practice in mathematics and helps to avoid errors.

Step 5: Final Answer

After going through all the steps, we have confidently arrived at the solution. We started with the equation 5(x - 4) = 3x + 4, distributed the 5, rearranged the equation to isolate x, and then verified our solution. Through these steps, we found that x = 12 is the correct answer. Therefore, the final answer to the equation 5(x - 4) = 3x + 4 is x = 12. This conclusive statement summarizes our entire problem-solving process and provides a clear and concise answer to the initial question. By presenting the final answer in this way, we ensure that the reader understands the result of our efforts and can confidently apply this knowledge to similar problems in the future. This final answer is the culmination of all our work and represents the solution we sought.

Conclusion

In conclusion, solving the equation 5(x - 4) = 3x + 4 involves several key steps, including distributing, rearranging, isolating the variable, and verifying the solution. We have demonstrated that the correct answer is x = 12. Understanding these steps is essential for mastering algebraic equations and building a strong foundation in mathematics. By breaking down the problem into manageable parts and providing clear explanations, we hope this guide has been helpful in understanding the process. This detailed approach not only helps in solving this particular equation but also equips readers with the skills to tackle a wide range of similar problems. Mathematics is a subject that builds upon itself, and mastering these fundamental concepts is crucial for success in more advanced topics. The ability to confidently solve equations is a valuable skill in both academic and practical settings. This conclusion reinforces the importance of the steps involved and the significance of the final answer in the context of algebraic problem-solving.