Solving 7x/2 - 5x/2 = 20x/3 + 10 A Step-by-Step Guide

Introduction to Linear Equations

In the realm of mathematics, solving equations is a fundamental skill. Linear equations, in particular, form the bedrock of many algebraic concepts. This article delves into the process of solving a specific linear equation: 7x/2 - 5x/2 = 20x/3 + 10. We will break down the steps involved, offering a clear and comprehensive guide suitable for students and anyone looking to refresh their algebra skills. Before we dive into the specifics of this equation, it is beneficial to understand the basic principles of solving linear equations. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. The goal in solving such an equation is to isolate the variable on one side of the equation, thereby determining its value. This involves performing the same operations on both sides of the equation to maintain balance. Common operations include addition, subtraction, multiplication, and division. Understanding these basics is crucial for tackling more complex equations and mathematical problems. Linear equations are not just abstract mathematical concepts; they have wide-ranging applications in various fields, from physics and engineering to economics and computer science. Mastering the techniques for solving linear equations opens doors to a deeper understanding of these disciplines. The principles we will discuss in the context of the given equation are universally applicable to any linear equation, making this a valuable exercise in algebraic problem-solving. As we proceed, we will emphasize clarity and accuracy, ensuring that each step is well-explained and easy to follow. This approach will not only help in solving the current equation but also build a strong foundation for future mathematical endeavors. So, let's embark on this journey to demystify the process of solving linear equations, one step at a time.

Step-by-Step Solution

To effectively solve the equation 7x/2 - 5x/2 = 20x/3 + 10, we will follow a structured, step-by-step approach. This method ensures clarity and reduces the likelihood of errors. Our initial focus will be on simplifying both sides of the equation independently before attempting to isolate the variable, x. This involves combining like terms and dealing with fractions to make the equation more manageable. The steps we take are crucial not only for obtaining the correct answer but also for developing a methodical approach to problem-solving in mathematics. This approach is transferable to other types of equations and mathematical problems, making it a valuable skill to acquire. Moreover, understanding the rationale behind each step enhances mathematical intuition and problem-solving abilities. Each step is justified by fundamental algebraic principles, which we will highlight as we proceed. This understanding is vital for building a strong foundation in algebra and mathematics in general. Our goal is not just to find the solution but to comprehend the process, enabling you to solve similar equations with confidence. The step-by-step format we adopt allows for a clear and logical progression, making the solution accessible to learners of all levels. So, let's begin our journey to solve this equation, carefully and methodically, one step at a time.

1. Combine Like Terms on the Left Side

Begin by focusing on the left side of the equation: 7x/2 - 5x/2. These terms share a common denominator, which makes combining them straightforward. The operation here involves subtracting the numerators while keeping the denominator constant. This is a fundamental principle in fraction arithmetic and is crucial for simplifying algebraic expressions. By combining like terms, we reduce the complexity of the equation, making it easier to manipulate and solve. This step aligns with the general strategy of simplifying equations before attempting to isolate the variable. Simplifying expressions early on reduces the chances of making errors in subsequent steps. Moreover, it often reveals underlying structures or patterns in the equation that might not be immediately apparent. This particular step highlights the importance of recognizing like terms in algebraic expressions and the rules for combining them. Proficiency in this area is essential for success in algebra and beyond. As we proceed, we will see how each step builds upon the previous one, leading us closer to the solution. Remember, the key to solving equations is to maintain balance and perform operations that simplify the expression while preserving equality. So, let's carry out this first step with precision and move on to the next stage of the solution. Understanding these basic algebraic manipulations is critical for mastering more advanced mathematical concepts.

2. Simplify the Left Side

After combining the terms, perform the subtraction in the numerator. This will further simplify the left side of the equation. The result will be a single term involving x, which is significantly easier to work with than the original expression. This simplification is a crucial step in isolating the variable. By reducing the number of terms, we make the equation more manageable and bring it closer to a form where we can directly solve for x. The process of simplifying algebraic expressions is a core skill in mathematics, and this step exemplifies its importance. It allows us to transform complex expressions into simpler, equivalent forms, making them easier to understand and manipulate. This step also reinforces the basic rules of arithmetic and their application in algebraic contexts. Accurate arithmetic is essential for obtaining the correct solution, so careful attention to detail is crucial. As we continue with the solution, we will see how each simplification brings us closer to our goal of finding the value of x. The ability to simplify expressions efficiently is a valuable asset in mathematics, and this step provides a clear example of how it can be applied in practice. So, let's complete this simplification and proceed to the next stage of the solution.

3. Eliminate the Fraction on the Left Side

Now that the left side is simplified, you may have a fraction involving x. To eliminate this fraction, multiply both sides of the equation by the denominator of the fraction. This is a fundamental algebraic technique for clearing fractions in equations. By multiplying both sides by the same value, we maintain the equality of the equation while eliminating the fraction, making the equation easier to solve. This step is often crucial in simplifying equations that involve fractions, as it allows us to work with whole numbers or simpler fractions instead. The principle behind this step is based on the multiplicative property of equality, which states that multiplying both sides of an equation by the same non-zero number preserves the equality. Understanding this principle is essential for solving equations correctly. Eliminating fractions often simplifies the subsequent steps, reducing the likelihood of errors and making the equation more accessible. This technique is widely used in algebra and is a valuable tool for solving various types of equations. As we proceed, we will see how this step contributes to the overall solution process, bringing us closer to isolating the variable. So, let's perform this multiplication and move on to the next step in solving the equation.

4. Eliminate the Fraction in The Whole Equation

If there are still fractions remaining in the equation, especially on the right side, identify the least common multiple (LCM) of the denominators of all fractions. Multiply both sides of the equation by this LCM. This step is crucial for clearing all fractions and simplifying the equation into a more manageable form. By multiplying by the LCM, we ensure that all denominators are canceled out, resulting in an equation with integer coefficients. This greatly simplifies the subsequent algebraic manipulations. The concept of the least common multiple is fundamental in number theory and is widely used in various mathematical contexts. Understanding how to find and use the LCM is essential for solving equations involving fractions. This step is based on the same principle as step 3, the multiplicative property of equality, but it is applied more broadly to clear all fractions in the equation. This technique is particularly useful when dealing with equations that have multiple fractions with different denominators. Eliminating fractions at this stage significantly reduces the complexity of the equation, making it easier to isolate the variable. As we continue with the solution, we will see how this step sets the stage for the final steps in solving for x. So, let's perform this multiplication carefully and move forward in our solution process.

5. Rearrange the Equation to Isolate x Terms

After eliminating fractions, the equation will likely have x terms on both sides. Rearrange the equation by adding or subtracting x terms to get all x terms on one side and constant terms on the other side. This step is a key part of isolating the variable, which is our ultimate goal in solving the equation. By grouping like terms together, we simplify the equation and bring it closer to a form where we can directly solve for x. This rearrangement is based on the additive property of equality, which states that adding or subtracting the same quantity from both sides of an equation preserves the equality. This property is fundamental in algebraic manipulations and is used extensively in solving equations. The specific operations required in this step will depend on the equation, but the principle remains the same: to group x terms on one side and constant terms on the other. This step often involves multiple additions or subtractions to achieve the desired arrangement. Careful attention to signs is crucial in this step to avoid errors. As we proceed, we will see how this rearrangement sets the stage for the final step in solving for x. So, let's perform these operations carefully and move forward in our solution process.

6. Solve for x

With all x terms on one side and constants on the other, you should have an equation in the form of ax = b. Divide both sides of the equation by the coefficient of x (which is 'a' in this case) to solve for x. This is the final step in isolating the variable and finding its value. This step is based on the multiplicative property of equality, specifically division, which states that dividing both sides of an equation by the same non-zero number preserves the equality. This property is fundamental in solving equations and is used extensively in algebra. The result of this division will be the value of x that satisfies the original equation. It is crucial to perform this division accurately to obtain the correct solution. Once we have solved for x, it is often a good practice to substitute the value back into the original equation to verify the solution. This ensures that the solution is correct and that no errors were made in the process. Solving for x is the culmination of all the previous steps, and it provides the answer to the equation. This step demonstrates the power of algebraic manipulation in solving problems. So, let's perform this final division and obtain the solution to our equation.

Conclusion

In conclusion, solving the equation 7x/2 - 5x/2 = 20x/3 + 10 involves a series of algebraic manipulations aimed at isolating the variable x. These steps, which include combining like terms, eliminating fractions, and rearranging the equation, are fundamental techniques in algebra. Mastering these techniques is crucial for success in mathematics and related fields. The process we have followed highlights the importance of a systematic approach to problem-solving. By breaking down the equation into smaller, manageable steps, we can solve it effectively and accurately. Each step is justified by basic algebraic principles, such as the additive and multiplicative properties of equality. Understanding these principles is essential for developing a strong foundation in algebra. Moreover, the techniques we have used are not specific to this particular equation; they can be applied to a wide range of linear equations and other algebraic problems. This makes the process we have described a valuable tool for anyone studying mathematics. The ability to solve equations is not just an academic skill; it has practical applications in various fields, from science and engineering to finance and economics. So, the effort invested in mastering these techniques is well worth it. We encourage you to practice solving similar equations to reinforce your understanding and build your confidence. With practice, you will become more proficient in algebraic problem-solving and better equipped to tackle more complex mathematical challenges. The journey of learning mathematics is a continuous one, and each equation solved is a step forward in that journey. Therefore, embrace the challenge, persevere through difficulties, and celebrate your successes along the way.