Solving (x+10)/(x-7) = 8/9 A Step-by-Step Guide
In this article, we will delve into the process of solving the equation (x+10)/(x-7) = 8/9. This is a typical algebraic equation involving fractions, and understanding how to solve it is crucial for anyone studying mathematics. We'll break down the steps in a clear, concise manner, ensuring that you grasp the underlying principles and can apply them to similar problems. This comprehensive guide will cover everything from the initial setup to the final solution, including explanations of the reasoning behind each step. Mastering such equations is vital for progressing in algebra and related fields.
Understanding the Basics of Algebraic Equations
Before we dive into solving the equation (x+10)/(x-7) = 8/9, it’s essential to grasp the fundamental concepts of algebraic equations. At its core, an algebraic equation is a mathematical statement asserting the equality of two expressions. These expressions can involve variables (like 'x' in our case), constants, and mathematical operations. The primary goal in solving an algebraic equation is to find the value(s) of the variable(s) that make the equation true. This involves isolating the variable on one side of the equation by performing the same operations on both sides, thereby maintaining the balance and integrity of the equation. For instance, if we add a number to one side, we must add the same number to the other side. Similarly, if we multiply one side by a factor, we must multiply the other side by the same factor. This principle of maintaining equality is the cornerstone of solving algebraic equations. In the context of our equation, (x+10)/(x-7) = 8/9, we need to manipulate the equation to isolate 'x'. This will involve steps such as cross-multiplication, combining like terms, and potentially other algebraic manipulations. The beauty of algebra lies in its systematic approach, allowing us to solve complex problems by breaking them down into simpler, manageable steps. This foundation is crucial not only for solving specific equations but also for understanding more advanced mathematical concepts.
Step-by-Step Solution of (x+10)/(x-7) = 8/9
Let's embark on a step-by-step journey to solve the equation (x+10)/(x-7) = 8/9. The first crucial step in tackling this equation is to eliminate the fractions. To achieve this, we employ a technique known as cross-multiplication. Cross-multiplication involves multiplying the numerator of the first fraction by the denominator of the second fraction, and vice versa. In our case, this translates to multiplying (x+10) by 9 and (x-7) by 8. This process effectively clears the denominators, transforming the equation into a more manageable form. The resulting equation will be 9(x+10) = 8(x-7). This step is pivotal because it simplifies the equation, making it easier to work with. By eliminating the fractions, we move closer to isolating 'x' and finding its value. Cross-multiplication is a standard technique in algebra, particularly useful when dealing with proportions or equations involving fractions. It allows us to convert a fractional equation into a linear equation, which is often simpler to solve. This initial step sets the stage for the subsequent algebraic manipulations that will lead us to the solution.
After performing the cross-multiplication, we arrive at the equation 9(x+10) = 8(x-7). The next step involves expanding both sides of the equation by applying the distributive property. The distributive property states that a(b + c) = ab + ac. Applying this to our equation, we multiply 9 by both 'x' and 10 on the left side, and 8 by both 'x' and -7 on the right side. This yields the equation 9x + 90 = 8x - 56. Expanding the equation is a crucial step as it removes the parentheses, making it easier to combine like terms and isolate 'x'. This process transforms the equation into a more simplified form, where the terms involving 'x' and the constant terms are clearly separated. The distributive property is a fundamental concept in algebra, and mastering its application is essential for solving equations and simplifying expressions. It allows us to deal with expressions enclosed in parentheses by distributing a factor across the terms inside. In our case, expanding the equation allows us to proceed with the next steps in solving for 'x'.
With the equation expanded to 9x + 90 = 8x - 56, the next critical step is to combine like terms. Combining like terms involves gathering the terms with the variable 'x' on one side of the equation and the constant terms on the other side. This is achieved by adding or subtracting terms from both sides of the equation to maintain equality. In our case, we can subtract 8x from both sides to get the 'x' terms on the left side, resulting in 9x - 8x + 90 = -56. Then, we can subtract 90 from both sides to move the constant terms to the right side, leading to 9x - 8x = -56 - 90. This process simplifies the equation by grouping similar terms together, making it easier to isolate 'x'. Combining like terms is a fundamental algebraic technique that streamlines the equation-solving process. It reduces the complexity of the equation by consolidating terms, which ultimately helps in finding the value of the variable. This step is essential for solving various types of algebraic equations and is a cornerstone of algebraic manipulation.
After combining like terms, our equation is now simplified to x = -146. This final step reveals the solution to the equation (x+10)/(x-7) = 8/9. The value of 'x' that satisfies the equation is -146. This means that if we substitute -146 for 'x' in the original equation, both sides will be equal. We have successfully isolated 'x' and found its value through a series of algebraic manipulations, including cross-multiplication, expansion using the distributive property, and combining like terms. This solution represents the value of 'x' that makes the equation a true statement. The process we followed demonstrates the systematic approach to solving algebraic equations, where each step is logically connected and aimed at isolating the variable. Understanding and applying these steps is crucial for solving a wide range of algebraic problems. The solution x = -146 is the culmination of our efforts and provides a definitive answer to the equation.
Verifying the Solution
Once we've obtained a solution, it's crucial to verify its correctness. This step ensures that the value we found for 'x' actually satisfies the original equation (x+10)/(x-7) = 8/9. To verify our solution, we substitute x = -146 back into the original equation. This involves replacing every instance of 'x' with -146 and then simplifying both sides of the equation to see if they are equal. Substituting -146 into the left side of the equation gives us (-146+10)/(-146-7). Simplifying the numerator gives us -136, and simplifying the denominator gives us -153. So, the left side becomes -136/-153, which simplifies to 136/153. Now, let's simplify this fraction further. Both 136 and 153 are divisible by 17. Dividing 136 by 17 gives us 8, and dividing 153 by 17 gives us 9. Therefore, the left side simplifies to 8/9. The right side of the original equation is already 8/9. Since both sides of the equation are equal when we substitute x = -146, we can confidently conclude that our solution is correct. This verification process is an essential part of problem-solving in mathematics. It provides a check against potential errors and ensures the accuracy of our solution.
Common Mistakes to Avoid
When solving algebraic equations like (x+10)/(x-7) = 8/9, it's easy to make mistakes if you're not careful. One common error is incorrect cross-multiplication. Remember, you need to multiply the numerator of the first fraction by the denominator of the second fraction, and vice versa. Forgetting to distribute the multiplication correctly is another frequent mistake. When you have an expression like 9(x+10), you must multiply both 'x' and 10 by 9. Similarly, with 8(x-7), both 'x' and -7 need to be multiplied by 8. Sign errors are also quite common, especially when dealing with negative numbers. Be very careful with your signs when combining like terms and performing arithmetic operations. Another pitfall is forgetting to verify your solution. Always substitute your answer back into the original equation to check if it's correct. This can help you catch any errors you might have made along the way. Skipping steps in the interest of speed can also lead to mistakes. It's better to write out each step clearly, especially when you're learning. By being aware of these common mistakes and taking steps to avoid them, you can improve your accuracy and confidence in solving algebraic equations.
Conclusion: Mastering Algebraic Equations
In conclusion, solving the equation (x+10)/(x-7) = 8/9 provides a valuable illustration of the techniques used to tackle algebraic equations. We've walked through the process step-by-step, from cross-multiplication to expanding expressions, combining like terms, and finally, isolating the variable 'x'. We arrived at the solution x = -146 and verified its correctness by substituting it back into the original equation. This process underscores the importance of a systematic approach in mathematics, where each step builds upon the previous one. We also highlighted common mistakes to avoid, such as incorrect distribution, sign errors, and the failure to verify the solution. Mastering these techniques is not just about solving specific equations; it's about developing a deeper understanding of algebraic principles that can be applied to a wide range of mathematical problems. Algebraic equations are a fundamental part of mathematics, and the ability to solve them is essential for success in higher-level math courses and various fields that rely on mathematical reasoning. By practicing these methods and being mindful of potential errors, you can enhance your problem-solving skills and build a solid foundation in algebra. The journey through this equation serves as a microcosm of the broader landscape of mathematical problem-solving, emphasizing the importance of precision, attention to detail, and a methodical approach.
