Solving (5x/2) + 7 = -3 A Step-by-Step Guide

Introduction

In the realm of mathematics, solving linear equations is a fundamental skill. Linear equations, characterized by a variable raised to the power of one, form the backbone of algebraic problem-solving. This article aims to provide a comprehensive, step-by-step guide on how to solve the linear equation ${\frac{5x}{2} + 7 = -3}$. By breaking down the process into manageable steps, we will empower you with the knowledge and confidence to tackle similar equations effectively. Our approach focuses on clarity and precision, ensuring that each step is explained in detail. Whether you are a student learning algebra for the first time or someone looking to refresh your skills, this guide will serve as a valuable resource. We will not only present the solution but also delve into the underlying principles that make this method work. Understanding these principles is crucial for long-term retention and the ability to apply these techniques to a broader range of problems. Remember, mathematics is not just about finding the right answer; it's about understanding the journey to that answer. So, let's embark on this mathematical journey together, unraveling the mystery of linear equations and enhancing your problem-solving abilities. Through this detailed explanation, you will gain a deeper appreciation for the elegance and power of algebra. The goal is not just to solve this particular equation, but to equip you with the tools and understanding to solve countless others. This equation, ${\frac{5x}{2} + 7 = -3}$, serves as a perfect example to illustrate the fundamental techniques used in solving linear equations. We will explore each step methodically, ensuring you grasp the logic and reasoning behind every operation. So, prepare to dive in and discover the step-by-step process of solving this linear equation.

Step 1: Isolate the Term with the Variable

Our primary goal in solving any equation is to isolate the variable, which in this case is 'x'. To do this effectively, we begin by isolating the term that contains the variable. In the equation ${\frac{5x}{2} + 7 = -3}$, the term with the variable is ${\frac{5x}{2}}$. The first step towards isolating this term is to eliminate the constant term, which is '+7' in this equation. To eliminate '+7', we apply the principle of inverse operations. This principle states that we can maintain the equality of an equation by performing the same operation on both sides. The inverse operation of addition is subtraction, so we subtract 7 from both sides of the equation. This gives us: ${\frac{5x}{2} + 7 - 7 = -3 - 7}$. By performing this subtraction, we effectively cancel out the '+7' on the left side of the equation, leaving us with the term containing the variable alone. Simplifying both sides, we get: ${\frac{5x}{2} = -10}$. This step is crucial because it brings us closer to isolating 'x' and solving for its value. Understanding the concept of inverse operations is fundamental in algebra, as it allows us to manipulate equations while preserving their balance. Each operation we perform on one side must be mirrored on the other side to maintain the equation's integrity. This step-by-step approach ensures that we methodically work towards the solution, minimizing errors and maximizing understanding. By isolating the variable term, we set the stage for the next steps in the solution process. This careful and deliberate approach is key to mastering algebraic problem-solving.

Step 2: Eliminate the Fraction

Having isolated the term with the variable, our next objective is to eliminate the fraction. In the equation ${\frac{5x}{2} = -10}$, the variable term is being divided by 2. To eliminate this fraction, we again employ the principle of inverse operations. The inverse operation of division is multiplication. Therefore, we multiply both sides of the equation by 2. This gives us: ${2 imes \frac{5x}{2} = -10 imes 2}$. Multiplying the left side by 2 cancels out the division by 2, effectively removing the fraction. On the right side, we multiply -10 by 2, which results in -20. Simplifying the equation, we now have: ${5x = -20}$. This step is significant because it simplifies the equation further, making it easier to solve for 'x'. Eliminating fractions is a common technique in algebra and is often a necessary step in solving equations. By multiplying both sides by the denominator of the fraction, we clear the fraction and obtain a simpler equation. This technique is not only applicable to this specific equation but can be used in a wide range of algebraic problems. Understanding how to manipulate fractions within equations is a crucial skill for any student of algebra. This step demonstrates the power of inverse operations in simplifying complex equations. By strategically applying the inverse operation, we move closer to isolating the variable and finding the solution. This methodical approach ensures that we proceed with accuracy and clarity, building a strong foundation for further algebraic problem-solving. The simplified equation, ${5x = -20}$, now sets the stage for the final step in solving for 'x'.

Step 3: Solve for x

With the equation simplified to ${5x = -20}$, the final step is to solve for x. The variable 'x' is currently being multiplied by 5. To isolate 'x', we once again utilize the principle of inverse operations. The inverse operation of multiplication is division. Therefore, we divide both sides of the equation by 5. This gives us: ${\frac{5x}{5} = \frac{-20}{5}}$. Dividing the left side by 5 cancels out the multiplication by 5, leaving 'x' isolated. On the right side, we divide -20 by 5, which results in -4. Simplifying the equation, we arrive at the solution: ${x = -4}$. This is the value of 'x' that satisfies the original equation, ${\frac{5x}{2} + 7 = -3}$. This final step demonstrates the culmination of all the previous steps. By systematically applying inverse operations, we have successfully isolated the variable and found its value. Solving for 'x' is the ultimate goal in this type of problem, and this step-by-step process ensures that we reach the solution accurately and efficiently. This process not only provides the answer but also reinforces the importance of understanding the underlying principles of algebra. The ability to solve for a variable is a fundamental skill in mathematics and has applications in various fields. By mastering this skill, you are equipping yourself with a powerful tool for problem-solving. The solution, ${x = -4}$, completes the process of solving the linear equation. This methodical approach, from isolating the variable term to eliminating fractions and finally solving for 'x', is a testament to the power of algebraic manipulation. Understanding and practicing these steps will build confidence and competence in solving linear equations.

Conclusion

In conclusion, we have successfully navigated the process of solving the linear equation ${\frac{5x}{2} + 7 = -3}$ using a clear, step-by-step approach. We began by isolating the term containing the variable, then eliminated the fraction, and finally solved for 'x'. The solution we arrived at is ${x = -4}$. This journey through the equation highlights the importance of understanding and applying inverse operations. Each step we took was deliberate and based on the fundamental principles of algebra. By subtracting 7 from both sides, we isolated the variable term. By multiplying both sides by 2, we eliminated the fraction. And by dividing both sides by 5, we solved for 'x'. This methodical approach is not only effective but also builds a strong foundation for solving more complex equations. The ability to solve linear equations is a cornerstone of mathematical proficiency. It is a skill that is widely applicable in various fields, from science and engineering to economics and finance. By mastering this skill, you are opening doors to a deeper understanding of the world around you. The process we have outlined is a template for solving a wide range of linear equations. While the specific numbers and operations may vary, the underlying principles remain the same. By practicing these steps and applying them to different equations, you will develop fluency and confidence in your problem-solving abilities. Remember, mathematics is not just about memorizing formulas; it's about understanding the logic and reasoning behind them. This step-by-step guide has aimed to provide not only the solution to this particular equation but also a deeper understanding of the process involved. We hope that this article has empowered you with the knowledge and skills to tackle linear equations with greater confidence. The journey of learning mathematics is a continuous one, and each equation you solve is a step forward in your mathematical journey. So, continue to explore, practice, and challenge yourself, and you will discover the beauty and power of mathematics.