Solving For X In (7x + 10) / (5x + 10) = 9/7 A Comprehensive Guide

Introduction

In this article, we will delve into the process of solving for x in the equation (7x + 10) / (5x + 10) = 9/7. This type of equation is a rational equation, where we have algebraic fractions set equal to each other. Solving such equations involves algebraic manipulation to isolate the variable x. This exploration is essential for anyone studying algebra or preparing for mathematical assessments. Understanding how to solve rational equations is crucial not only for academic success but also for real-world problem-solving scenarios where mathematical models are used. The ability to manipulate and solve equations is a foundational skill in mathematics, and this particular equation provides a comprehensive example of the techniques involved. The process will involve several key steps, including cross-multiplication, simplification, and isolating the variable. Let's dive into a detailed step-by-step solution to ensure a clear understanding of the methodology. We will also discuss common pitfalls to avoid and strategies to verify the solution, making this guide a valuable resource for learners of all levels.

Step-by-Step Solution

The given equation is:

$$\frac{7x + 10}{5x + 10} = \frac{9}{7}$$

The first step in solving this equation is to eliminate the fractions by cross-multiplying. This involves multiplying the numerator of the left side by the denominator of the right side, and vice versa. This technique is based on the fundamental property of proportions, which states that if a/b = c/d, then ad = bc. Applying this to our equation:

$$7(7x + 10) = 9(5x + 10)$$

Now, we need to distribute the constants on both sides of the equation. This means multiplying the number outside the parentheses by each term inside the parentheses. This is a crucial step in simplifying the equation and preparing it for further manipulation. The distributive property, a(b + c) = ab + ac, is the backbone of this step. Applying this property carefully ensures that we expand the equation correctly, maintaining the equality throughout the process. Here’s how we do it:

$$49x + 70 = 45x + 90$$

Next, we want to collect like terms. This involves bringing all the terms with x to one side of the equation and the constant terms to the other side. This step is essential for isolating the variable x. To do this, we subtract 45x from both sides and subtract 70 from both sides. This maintains the balance of the equation while organizing the terms in a way that allows us to solve for x. Here’s the process:

$$49x - 45x = 90 - 70$$

Now, simplify both sides by combining like terms:

$$4x = 20$$

Finally, to isolate x, we divide both sides of the equation by the coefficient of x, which is 4:

$$x = \frac{20}{4}$$

$$x = 5$$

Thus, the solution to the equation (7x + 10) / (5x + 10) = 9/7 is x = 5. This step completes the algebraic manipulation, giving us the value of x that satisfies the original equation. The final solution is the culmination of several carefully executed steps, each building upon the previous one to progressively isolate the variable. This process underscores the importance of precision and attention to detail in algebra. The solution x = 5 is not just a number; it’s the value that, when substituted back into the original equation, will make the two sides equal. This is a critical concept in solving equations, and it's why we often verify our solutions to ensure accuracy.

Verification

To ensure our solution is correct, we substitute x = 5 back into the original equation:

$$\frac{7(5) + 10}{5(5) + 10} = \frac{9}{7}$$

Simplify the numerator and the denominator:

$$\frac{35 + 10}{25 + 10} = \frac{9}{7}$$

$$\frac{45}{35} = \frac{9}{7}$$

Reduce the fraction on the left side by dividing both the numerator and the denominator by their greatest common divisor, which is 5:

$$\frac{9}{7} = \frac{9}{7}$$

Since the left side equals the right side, our solution x = 5 is correct. This verification step is crucial in mathematics, as it confirms that the algebraic manipulations have been performed correctly and that the solution satisfies the initial equation. It provides a check against errors that may have occurred during the solving process, ensuring the reliability of the result. By substituting the value back into the original equation, we are essentially reversing the process to see if we arrive back at the starting point, confirming the validity of our solution.

Common Mistakes to Avoid

When solving equations, especially rational equations like the one we've discussed, several common mistakes can occur. Being aware of these pitfalls can help you avoid errors and arrive at the correct solution more efficiently. Here are some of the most frequent mistakes:

1. Incorrect Distribution: One of the most common errors is failing to distribute correctly when multiplying a constant by an expression in parentheses. For example, in the step 7(7x + 10) = 49x + 70, students might incorrectly write 7(7x + 10) = 49x + 10, forgetting to multiply 7 by both terms inside the parentheses. Double-checking each term after distribution can help avoid this mistake. This error can significantly alter the outcome, leading to an incorrect solution. Accurate distribution is vital for maintaining the equation's balance and integrity.

2. Sign Errors: Sign errors often occur when moving terms from one side of the equation to the other. For instance, when we subtract 45x from both sides, it's essential to ensure the sign changes correctly. A mistake like not changing the sign can lead to an incorrect solution. It's a good practice to rewrite the equation after each term movement to visually confirm the changes. Sign errors are particularly tricky because they can subtly throw off the calculation without being immediately apparent.

3. Dividing Incorrectly: When isolating x, ensure you divide both sides of the equation by the correct coefficient. For instance, in the equation 4x = 20, dividing only one side by 4 will lead to an incorrect solution. It's crucial to perform the same operation on both sides to maintain equality. This principle is a cornerstone of algebraic manipulation, and overlooking it can invalidate the entire solution process. Double-checking this step can save time and prevent frustration.

4. Forgetting to Verify: As we demonstrated earlier, verifying the solution by substituting it back into the original equation is a crucial step. Many students skip this step, which can lead to accepting an incorrect solution. Verification is the ultimate check on your work and ensures the reliability of your answer. It not only confirms the solution's accuracy but also reinforces your understanding of the problem-solving process.

5. Not Simplifying Fractions Correctly: When dealing with rational equations, simplifying fractions correctly is essential. For example, if you end up with a fraction like 45/35, you must simplify it to 9/7. Failing to do so can complicate the equation unnecessarily and lead to errors. Always look for common factors between the numerator and the denominator and simplify them to their lowest terms. Simplification not only makes the numbers easier to work with but also helps in recognizing potential simplifications and patterns in the equation.

By being mindful of these common mistakes, you can improve your accuracy and confidence in solving algebraic equations. Each mistake highlights a key aspect of algebraic manipulation, reinforcing the need for careful attention to detail and a systematic approach. Consistent practice and self-checking are the best ways to avoid these errors and develop strong problem-solving skills.

Real-World Applications

The ability to solve equations like (7x + 10) / (5x + 10) = 9/7 isn't just a theoretical exercise; it has numerous real-world applications. Algebraic equations are the foundation of many fields, including engineering, physics, economics, and computer science. Understanding how to manipulate and solve them is crucial for tackling practical problems.

In engineering, equations are used to design structures, calculate loads, and analyze systems. For example, engineers might use equations to determine the optimal dimensions of a bridge or the flow rate in a pipe system. The principles of algebra help engineers to model physical phenomena and design solutions that meet specific requirements. Whether it's designing a circuit or planning a building, equations are the language of engineering.

In physics, equations are used to describe the motion of objects, the behavior of electricity and magnetism, and the properties of light and sound. Solving equations allows physicists to make predictions about the physical world and to test their theories through experiments. From calculating the trajectory of a projectile to understanding the behavior of subatomic particles, equations are indispensable tools for physicists. They provide a quantitative framework for understanding the fundamental laws of nature.

Economics uses equations to model supply and demand, predict market behavior, and analyze economic trends. Economists might use equations to forecast inflation, calculate interest rates, or determine the impact of government policies on the economy. These models help economists and policymakers make informed decisions and understand the complex interactions within economic systems. The use of algebraic equations in economics provides a structured approach to analyzing and predicting economic outcomes.

In computer science, equations are used in algorithm design, data analysis, and machine learning. Computer scientists rely on equations to optimize algorithms, develop statistical models, and build intelligent systems. From writing code to designing databases, equations provide a foundation for computational thinking. The ability to solve and manipulate equations is essential for creating efficient and effective software and systems.

Moreover, the skills you develop in solving equations also translate to everyday problem-solving. Whether you're calculating a budget, determining the best deal on a purchase, or planning a project, the logical thinking and analytical skills you hone in algebra are invaluable. The ability to break down a problem into smaller parts, identify the key variables, and formulate an equation to solve it is a versatile skill that benefits you in many areas of life. This connection between algebraic skills and real-world applications underscores the importance of mastering these concepts.

Conclusion

Solving the equation (7x + 10) / (5x + 10) = 9/7 demonstrates the fundamental principles of algebra and the importance of careful manipulation. By following a step-by-step approach, we successfully found that x = 5. Remember to always verify your solution to ensure accuracy. The skills acquired in solving this type of equation are transferable to many other areas of mathematics and have practical applications in various real-world scenarios. The process of solving equations is not just about finding an answer; it's about developing a systematic approach to problem-solving. This approach includes understanding the underlying principles, performing manipulations with precision, and verifying the results. Each step in the process reinforces the importance of attention to detail and logical thinking. The ability to solve algebraic equations is a foundational skill that opens doors to further study in mathematics and related fields. It provides a framework for understanding more complex concepts and tackling more challenging problems. Mastering these skills enhances your analytical abilities and prepares you for success in a wide range of academic and professional pursuits. Therefore, investing time and effort in understanding and practicing these techniques is a worthwhile endeavor. The benefits extend far beyond the classroom, equipping you with valuable tools for problem-solving in various aspects of life.