Solving For X In 7x - 1 = 5x + 5 A Step-by-Step Guide
Introduction
In the realm of algebra, solving for x in a linear equation is a fundamental skill. Linear equations, characterized by a highest power of 1 for the variable, appear ubiquitously in mathematics, science, and engineering. Mastering the technique to solve these equations is crucial for tackling more complex problems. This article provides a detailed walkthrough of solving the equation 7x - 1 = 5x + 5, explaining each step in a clear and concise manner. We will delve into the underlying principles of algebraic manipulation, ensuring that you not only understand the solution but also the rationale behind it. Whether you are a student grappling with algebra for the first time or someone looking to brush up on your skills, this guide offers valuable insights and practical advice. This initial introduction will set the stage for a thorough exploration of linear equation-solving strategies, providing a solid foundation for further mathematical endeavors. Understanding how to isolate the variable x is not merely about finding a numerical answer; it's about developing a logical and systematic approach to problem-solving that can be applied across various disciplines. The step-by-step methodology discussed herein will empower you to confidently tackle similar equations, fostering a deeper appreciation for the elegance and precision of algebraic techniques. The ability to manipulate equations and isolate variables is a cornerstone of mathematical literacy, enabling you to analyze relationships, make predictions, and solve real-world problems. So, let's embark on this journey of unraveling the equation 7x - 1 = 5x + 5, and in the process, enhance your algebraic prowess.
Understanding Linear Equations
Before diving into the solution, it's essential to grasp the concept of linear equations. A linear equation is an algebraic equation where each term is either a constant or the product of a constant and a single variable. The variable in a linear equation is raised to the power of one, and the equation, when graphed, represents a straight line. Examples of linear equations include 2x + 3 = 7, y = mx + b, and the equation we are addressing today, 7x - 1 = 5x + 5. Understanding the structure and properties of linear equations is crucial for efficiently solving them. The goal in solving a linear equation is to isolate the variable on one side of the equation, thereby determining its value. This isolation is achieved by performing the same operations on both sides of the equation, maintaining the balance and preserving the equality. The fundamental principle underlying this approach is that any operation performed on one side of the equation must also be performed on the other side to keep the equation valid. This principle ensures that the solution obtained is accurate and consistent with the original equation. Furthermore, recognizing the linear nature of an equation allows us to apply specific techniques tailored to this type of equation, such as combining like terms and using inverse operations. By mastering the properties of linear equations, you'll be well-equipped to solve a wide range of algebraic problems. The ability to identify and manipulate these equations forms the bedrock of more advanced mathematical concepts, such as systems of equations and linear transformations. Therefore, a solid understanding of linear equations is not only essential for solving equations but also for building a strong foundation in mathematics as a whole.
Step-by-Step Solution for 7x - 1 = 5x + 5
Now, let's dissect the solution for the equation 7x - 1 = 5x + 5 step-by-step. The primary objective is to isolate x on one side of the equation. This involves a series of algebraic manipulations that preserve the equality. The first step is to gather all terms containing x on one side and all constant terms on the other. A common approach is to subtract the smaller x term from both sides. In this case, we subtract 5x from both sides of the equation: 7x - 1 - 5x = 5x + 5 - 5x. This simplifies to 2x - 1 = 5. The next step is to isolate the x term by removing the constant term on the same side. We achieve this by adding 1 to both sides of the equation: 2x - 1 + 1 = 5 + 1. This simplifies to 2x = 6. Finally, to solve for x, we divide both sides of the equation by the coefficient of x, which is 2: 2x / 2 = 6 / 2. This gives us the solution x = 3. Each step in this process is crucial, ensuring that the equation remains balanced and the solution accurate. By meticulously applying these steps, we successfully isolated x and determined its value. This methodical approach highlights the importance of precision and attention to detail in algebraic problem-solving. Furthermore, understanding the rationale behind each step is as important as arriving at the correct answer. It allows you to adapt these techniques to solve other linear equations with confidence and proficiency. So, let's recap the key steps: subtract 5x from both sides, add 1 to both sides, and divide both sides by 2. This sequence of operations forms the cornerstone of solving linear equations of this type.
Step 1: Subtract 5x from both sides
Our initial equation is 7x - 1 = 5x + 5. To begin isolating x, we need to consolidate the x terms on one side of the equation. A strategic move is to subtract the smaller x term, which is 5x, from both sides. This maintains the equation's balance while moving us closer to isolating x. The equation becomes: 7x - 1 - 5x = 5x + 5 - 5x. This subtraction operation is a fundamental algebraic principle, ensuring that if we perform the same operation on both sides of an equation, the equality remains valid. This is analogous to a balanced scale – if you remove the same weight from both sides, the scale remains balanced. The subtraction of 5x from both sides serves a dual purpose: it eliminates the x term from the right side of the equation and simplifies the left side by combining like terms. The like terms on the left side are 7x and -5x, which combine to give 2x. Therefore, the equation simplifies to 2x - 1 = 5. This step is crucial because it reduces the complexity of the equation, making it easier to solve for x. The act of subtracting 5x from both sides demonstrates a key aspect of algebraic manipulation: the ability to strategically alter the equation's form without changing its fundamental meaning. By carefully choosing this operation, we have successfully streamlined the equation and brought it closer to a form where x can be easily isolated. This methodical approach is a hallmark of effective problem-solving in algebra.
Step 2: Add 1 to both sides
After subtracting 5x from both sides, our equation stands as 2x - 1 = 5. The next crucial step in isolating x is to eliminate the constant term, -1, from the left side. To achieve this, we employ the inverse operation of subtraction, which is addition. By adding 1 to both sides of the equation, we maintain the balance and move closer to our goal. The equation transforms as follows: 2x - 1 + 1 = 5 + 1. This addition operation, like the subtraction in the previous step, is a cornerstone of algebraic manipulation. It ensures that the equality of the equation is preserved while allowing us to strategically modify its form. Adding 1 to both sides is a targeted action aimed at canceling out the -1 on the left side. The terms -1 and +1 are additive inverses, meaning they sum to zero. This cancellation simplifies the left side of the equation, leaving us with just the term containing x. On the right side, the addition is straightforward: 5 + 1 = 6. Therefore, after adding 1 to both sides, the equation simplifies to 2x = 6. This simplified form is significantly easier to solve than the original equation. It highlights the power of using inverse operations to isolate variables in algebraic equations. The act of adding 1 to both sides is not merely an arithmetic operation; it's a strategic maneuver that propels us towards the solution. This step demonstrates the importance of recognizing the inverse relationships between operations in algebra and leveraging them to simplify equations.
Step 3: Divide both sides by 2
Having simplified the equation to 2x = 6, the final step in solving for x involves isolating the variable completely. The term 2x represents 2 multiplied by x. To undo this multiplication, we employ the inverse operation, which is division. By dividing both sides of the equation by 2, we isolate x and determine its value. The equation transforms as follows: 2x / 2 = 6 / 2. This division operation is the culmination of our algebraic manipulations, bringing us to the solution. It's essential to remember that dividing both sides of an equation by the same non-zero number maintains the equality. This principle is fundamental to solving algebraic equations. On the left side, dividing 2x by 2 effectively cancels out the 2, leaving us with just x. This is the isolation we have been working towards. On the right side, the division is straightforward: 6 / 2 = 3. Therefore, after dividing both sides by 2, we arrive at the solution: x = 3. This final step underscores the importance of inverse operations in algebra. By recognizing the multiplicative relationship between 2 and x and applying the inverse operation of division, we successfully isolated x. The solution x = 3 represents the value that, when substituted back into the original equation 7x - 1 = 5x + 5, makes the equation true. This verification is a crucial step in problem-solving, ensuring the accuracy of our solution. The division operation is not just a numerical calculation; it's the key to unlocking the value of x. This step-by-step process, from subtracting 5x to adding 1 and finally dividing by 2, exemplifies the systematic approach required to solve linear equations effectively.
Verification of the Solution
Once we've arrived at a solution, it's crucial to verify its accuracy. This process involves substituting the value we found for x back into the original equation and checking if the equation holds true. In our case, we found that x = 3 for the equation 7x - 1 = 5x + 5. To verify this, we substitute 3 for x in the original equation: 7(3) - 1 = 5(3) + 5. Now, we simplify both sides of the equation separately. On the left side, 7(3) - 1 becomes 21 - 1, which equals 20. On the right side, 5(3) + 5 becomes 15 + 5, which also equals 20. Since both sides of the equation simplify to the same value, 20 = 20, the equation holds true. This confirms that our solution, x = 3, is correct. Verification is not merely a formality; it's an essential step in problem-solving. It provides assurance that the algebraic manipulations we performed were accurate and that we arrived at the correct value for the variable. This process is particularly important in more complex equations where errors can easily occur. By verifying the solution, we can catch any mistakes and correct them before moving on. Furthermore, verification reinforces our understanding of the equation and the solution process. It solidifies the concept that the value we found for x satisfies the equation's conditions. This practice of verifying solutions builds confidence in our problem-solving abilities and promotes a meticulous approach to mathematics. So, remember to always verify your solutions, ensuring accuracy and enhancing your understanding of the underlying concepts.
Conclusion
In conclusion, solving for x in the equation 7x - 1 = 5x + 5 exemplifies the fundamental principles of algebraic manipulation. By systematically applying inverse operations and maintaining the equation's balance, we successfully isolated x and determined its value to be 3. The process involved subtracting 5x from both sides, adding 1 to both sides, and finally, dividing both sides by 2. Each step was crucial in simplifying the equation and bringing us closer to the solution. Furthermore, we emphasized the importance of verifying the solution by substituting the value back into the original equation, ensuring its accuracy. This verification step is a cornerstone of mathematical problem-solving, providing confidence in the correctness of our answer. The skills and techniques demonstrated in this guide are applicable to a wide range of linear equations and form the foundation for more advanced algebraic concepts. Mastering these techniques not only enables you to solve equations but also fosters critical thinking and problem-solving skills that are valuable in various aspects of life. The ability to manipulate equations, isolate variables, and verify solutions is a testament to your mathematical proficiency. So, embrace the challenges of algebra, practice these techniques, and continue to build your mathematical prowess. The journey of learning mathematics is a continuous one, and each equation solved is a step forward in your understanding and appreciation of this powerful discipline.
