Solving Algebraic Equations Step-by-Step Find X In 7(4x-1) + X = 7x + 3
Introduction
In this article, we will delve into the process of solving for the variable x in the algebraic equation 7(4x-1) + x = 7x + 3. This equation involves the use of the distributive property, combining like terms, and isolating the variable to find its value. Algebraic equations like this one are fundamental in mathematics and have applications in various fields, including physics, engineering, and economics. Understanding how to solve them is a crucial skill for anyone studying these disciplines. The steps involved in solving this equation illustrate core algebraic principles that can be applied to a wide range of problems. We will break down each step in detail, ensuring clarity and understanding. By the end of this article, you will have a solid grasp of how to solve similar equations and be able to apply these techniques to more complex problems. This skill is not only useful in academic settings but also in real-world scenarios where problem-solving and analytical thinking are essential. Let’s embark on this mathematical journey and unravel the value of x.
Understanding the Equation
The equation we aim to solve is 7(4x-1) + x = 7x + 3. This equation combines several algebraic concepts, including the distributive property, combining like terms, and isolating the variable. Before we begin solving, it's crucial to understand each component. The term 7(4x-1) indicates that the number 7 is multiplied by the entire expression within the parentheses. This requires the use of the distributive property, which we will explore in detail later. The equation also includes terms with the variable x and constant terms (numbers without variables). To solve for x, we need to combine like terms, meaning we group together terms that have the same variable and constant terms together. The ultimate goal is to isolate x on one side of the equation, which means we want to get x by itself. This is achieved by performing inverse operations, such as adding or subtracting terms from both sides of the equation. By understanding these basic principles, we can approach the equation systematically and confidently. This section lays the groundwork for the step-by-step solution that follows, ensuring that you have a clear understanding of the underlying concepts.
Step-by-Step Solution
1. Apply the Distributive Property
Our first step in solving the equation 7(4x-1) + x = 7x + 3 is to apply the distributive property. The distributive property states that a(b + c) = ab + ac. In our case, we need to distribute the 7 across the terms inside the parentheses (4x-1). This means we multiply 7 by 4x and 7 by -1. So, 7 * 4x equals 28x, and 7 * -1 equals -7. Applying the distributive property transforms the left side of the equation from 7(4x-1) + x to 28x - 7 + x. This step is crucial because it removes the parentheses, allowing us to combine like terms in the next step. The distributive property is a fundamental concept in algebra, and mastering its application is essential for solving a wide range of equations. By correctly applying this property, we simplify the equation and bring it closer to a form where we can isolate the variable x.
2. Combine Like Terms
After applying the distributive property, our equation looks like this: 28x - 7 + x = 7x + 3. The next step is to combine like terms on each side of the equation. Like terms are terms that have the same variable raised to the same power, or constant terms (numbers without variables). On the left side of the equation, we have two terms with the variable x: 28x and x. Combining these terms means adding their coefficients (the numbers in front of the variables). So, 28x + x equals 29x. The constant term on the left side is -7, which remains as it is since there are no other constant terms to combine it with. Therefore, the left side of the equation simplifies to 29x - 7. On the right side of the equation, we have 7x + 3. There are no like terms to combine on this side, so it remains as it is. Our equation now looks like this: 29x - 7 = 7x + 3. This step is essential because it simplifies the equation, making it easier to isolate the variable x. Combining like terms is a fundamental algebraic skill that helps in reducing complex expressions to simpler forms.
3. Isolate the Variable
Now that we have simplified the equation to 29x - 7 = 7x + 3, our next goal is to isolate the variable x on one side of the equation. This involves a series of steps to move all the x terms to one side and all the constant terms to the other side. First, let's subtract 7x from both sides of the equation. This will eliminate the x term on the right side. Subtracting 7x from both sides gives us: 29x - 7x - 7 = 7x - 7x + 3. This simplifies to 22x - 7 = 3. Next, we need to move the constant term -7 from the left side to the right side. To do this, we add 7 to both sides of the equation: 22x - 7 + 7 = 3 + 7. This simplifies to 22x = 10. We have now successfully isolated the x term on the left side of the equation. This step is crucial because it brings us closer to finding the value of x. By performing these inverse operations (subtraction and addition), we maintain the balance of the equation while moving terms to their appropriate sides. Isolating the variable is a key technique in solving algebraic equations, and it demonstrates a strong understanding of algebraic principles.
4. Solve for x
After isolating the variable, we have the equation 22x = 10. The final step in solving for x is to divide both sides of the equation by the coefficient of x, which is 22. This will give us the value of x. Dividing both sides by 22, we get: (22x) / 22 = 10 / 22. This simplifies to x = 10 / 22. Now, we can simplify the fraction 10 / 22 by dividing both the numerator and the denominator by their greatest common divisor, which is 2. So, 10 / 2 = 5, and 22 / 2 = 11. Therefore, the simplified fraction is 5 / 11. Thus, the solution to the equation is x = 5 / 11. This is the value of x that satisfies the original equation. This step is the culmination of all the previous steps, where we use the principles of algebra to find the numerical value of the variable. By dividing both sides of the equation by the coefficient of x, we effectively isolate x and determine its value. This process demonstrates a thorough understanding of algebraic manipulation and problem-solving skills.
Verification
To ensure that our solution x = 5 / 11 is correct, it's crucial to verify it by substituting this value back into the original equation: 7(4x-1) + x = 7x + 3. Substitute x = 5 / 11 into the equation: 7(4(5 / 11)-1) + (5 / 11) = 7(5 / 11) + 3. First, let's simplify the expression inside the parentheses: 4(5 / 11) = 20 / 11. So, 4(5 / 11) - 1 = (20 / 11) - 1. To subtract 1 from 20 / 11, we need to express 1 as a fraction with a denominator of 11, which is 11 / 11. Therefore, (20 / 11) - (11 / 11) = 9 / 11. Now, substitute this back into the equation: 7(9 / 11) + (5 / 11) = 7(5 / 11) + 3. Next, multiply 7 by 9 / 11: 7 * (9 / 11) = 63 / 11. So, the left side of the equation becomes: (63 / 11) + (5 / 11). Adding these fractions gives us: (63 + 5) / 11 = 68 / 11. Now, let's simplify the right side of the equation: 7 * (5 / 11) = 35 / 11. So, the right side of the equation becomes: (35 / 11) + 3. To add 3 to 35 / 11, we need to express 3 as a fraction with a denominator of 11, which is 33 / 11. Therefore, (35 / 11) + (33 / 11) = (35 + 33) / 11 = 68 / 11. Comparing both sides of the equation, we have: 68 / 11 = 68 / 11. Since both sides are equal, our solution x = 5 / 11 is verified to be correct. This verification step is an essential part of the problem-solving process, as it confirms the accuracy of our solution and gives us confidence in our result.
Common Mistakes to Avoid
When solving algebraic equations, there are several common mistakes that students often make. Avoiding these mistakes can significantly improve your accuracy and problem-solving skills. One common mistake is an incorrect application of the distributive property. Remember that the distributive property requires multiplying the term outside the parentheses by each term inside the parentheses. For example, in the equation 7(4x-1) + x = 7x + 3, you must multiply 7 by both 4x and -1. A mistake here would lead to an incorrect equation. Another common error is failing to combine like terms correctly. Make sure to only combine terms that have the same variable raised to the same power, or constant terms. For instance, in the expression 28x - 7 + x, you should combine 28x and x to get 29x, but you cannot combine 29x with -7. Incorrectly combining terms will lead to a wrong simplification of the equation. A further mistake is not performing the same operation on both sides of the equation. When isolating the variable, it's crucial to maintain the balance of the equation. If you add or subtract a term on one side, you must do the same on the other side. For example, if you subtract 7x from the left side, you must also subtract 7x from the right side. Failing to do so will disrupt the equality and lead to an incorrect solution. Sign errors are also frequent, especially when dealing with negative numbers. Pay close attention to the signs of the terms and ensure you are applying the correct operations. For instance, when moving a term from one side of the equation to the other, remember to change its sign. By being mindful of these common mistakes, you can enhance your problem-solving accuracy and approach algebraic equations with greater confidence. Double-checking each step and verifying your solution are also good practices to adopt.
Conclusion
In summary, solving the equation 7(4x-1) + x = 7x + 3 involves several key algebraic steps. We began by applying the distributive property to remove the parentheses, which gave us 28x - 7 + x = 7x + 3. Next, we combined like terms on each side of the equation, simplifying it to 29x - 7 = 7x + 3. Then, we isolated the variable x by subtracting 7x from both sides and adding 7 to both sides, resulting in 22x = 10. Finally, we solved for x by dividing both sides by 22, which yielded x = 5 / 11. We also verified our solution by substituting x = 5 / 11 back into the original equation, confirming its correctness. Throughout this process, we highlighted the importance of each step and the underlying algebraic principles involved. We also discussed common mistakes to avoid, such as incorrect application of the distributive property, failing to combine like terms correctly, and not performing the same operation on both sides of the equation. Mastering these techniques and avoiding common errors are crucial for success in algebra and beyond. Algebraic problem-solving is not only a fundamental skill in mathematics but also a valuable tool in various fields that require analytical thinking and logical reasoning. By understanding and practicing these steps, you can confidently tackle similar equations and enhance your mathematical abilities. The ability to solve algebraic equations is a stepping stone to more advanced mathematical concepts and real-world applications.
