Solving The Equation 5/(4x-9) = -5 A Step-by-Step Guide
In the realm of mathematics, solving equations is a fundamental skill. It allows us to find the value(s) of unknown variables that satisfy a given relationship. This article dives deep into the process of solving the equation 5/(4x - 9) = -5, providing a detailed, step-by-step guide suitable for both beginners and those looking to refresh their algebraic skills. We'll explore the underlying concepts, break down the solution process, and discuss potential pitfalls to avoid, ensuring a comprehensive understanding of this essential mathematical technique. The ability to solve equations is not just a skill confined to the classroom; it's a crucial tool in various fields, including science, engineering, economics, and even everyday problem-solving. By mastering these techniques, you empower yourself to tackle a wide range of challenges and make informed decisions. So, let's embark on this journey of algebraic exploration and unravel the solution to the equation at hand.
At its core, solving an equation involves isolating the unknown variable on one side of the equation. This is achieved by performing algebraic operations that maintain the equality, such as adding, subtracting, multiplying, or dividing both sides by the same quantity. The key is to apply these operations strategically, working towards simplifying the equation until the variable stands alone. This may involve combining like terms, distributing coefficients, and undoing operations to gradually peel away the layers surrounding the variable. In more complex equations, techniques like factoring, completing the square, or using the quadratic formula might be necessary. However, the fundamental principle remains the same: to manipulate the equation in a way that reveals the value(s) of the unknown variable(s). Understanding the properties of equality and the order of operations is crucial for success in solving equations. With practice and a systematic approach, you can confidently tackle a wide variety of algebraic challenges.
Before diving into the specifics of the given equation, it's helpful to recap some of the key concepts and principles that underpin equation solving. Remember that an equation is a statement of equality between two expressions. The goal is to find the value(s) of the variable(s) that make this statement true. Each algebraic operation you perform on one side of the equation must be mirrored on the other side to maintain the balance. Think of the equation as a scale – if you add weight to one side, you must add the same weight to the other side to keep it level. This fundamental principle ensures that the solution you find is valid. The order of operations (PEMDAS/BODMAS) also plays a crucial role. When simplifying expressions within an equation, follow the correct order: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). This ensures consistency and accuracy in your calculations. Finally, be mindful of potential pitfalls, such as dividing by zero (which is undefined) or incorrectly applying the distributive property. A solid grasp of these foundational concepts will set you up for success in solving equations of all types.
Step 1: Eliminate the Fraction
The given equation is: $ rac{5}{4x-9} = -5$. Our first goal is to eliminate the fraction to simplify the equation. We can achieve this by multiplying both sides of the equation by the denominator, which is . This step is crucial because it removes the fraction, making the equation easier to manipulate. Multiplying both sides by the same expression maintains the equality, a fundamental principle in solving equations. It's like multiplying both sides of a balanced scale by the same weight – the scale remains balanced. Be careful to distribute any multiplication correctly to avoid errors. This initial step sets the stage for further simplification and ultimately leads us to the solution. By eliminating the fraction, we transform the equation into a more manageable form, paving the way for isolating the variable x. Remember, the key to successful equation solving is to systematically simplify the equation, one step at a time, until the variable is alone on one side. This step-by-step approach minimizes the risk of errors and allows for a clear and logical progression towards the solution.
When multiplying both sides by , we get: $5 = -5(4x - 9)$. This equation is now free of fractions, making it easier to work with. This transformation is a key step in solving equations involving fractions. By eliminating the denominator, we've essentially cleared a hurdle and simplified the problem significantly. It's like removing an obstacle from a path, allowing for smoother progress. Now, the equation is in a form where we can apply other algebraic techniques, such as the distributive property, to further simplify and isolate the variable x. This process of transforming an equation into a more manageable form is a common strategy in algebra. By strategically applying algebraic operations, we can gradually peel away the layers surrounding the variable, bringing us closer to the solution. The ability to recognize and apply these simplifying techniques is a hallmark of strong algebraic skills. So, let's continue our journey, armed with this simplified equation, and move towards finding the value of x.
It's important to note a potential pitfall at this stage: we must ensure that the value of x we ultimately find does not make the original denominator, , equal to zero. If it does, then the original equation would be undefined, as division by zero is not allowed. This is a crucial consideration when dealing with equations involving fractions. We need to check our solution against this condition to ensure its validity. In this case, we need to make sure that . This translates to x e rac{9}{4}. So, if our final solution is rac{9}{4}, we would need to discard it and conclude that the equation has no solution. This step of checking for extraneous solutions is an essential part of the equation-solving process. It ensures that we only accept solutions that are valid within the context of the original equation. Failing to check for extraneous solutions can lead to incorrect answers, so it's a habit worth cultivating. With this caveat in mind, we can confidently proceed with solving the simplified equation, knowing that we will need to verify our solution at the end.
Step 2: Distribute on the Right Side
Next, we need to distribute the -5 on the right side of the equation: $5 = -5(4x - 9)$. Applying the distributive property, we multiply -5 by both terms inside the parentheses: $5 = -20x + 45$. The distributive property is a fundamental concept in algebra that allows us to simplify expressions involving parentheses. It states that a(b + c) = ab + ac. In this case, -5 is multiplied by both 4x and -9, resulting in -20x and +45, respectively. The correct application of the distributive property is crucial for accurate equation solving. A common mistake is to only multiply the term outside the parentheses by the first term inside, neglecting the second term. So, always remember to multiply the term outside by each term inside the parentheses. This step of distributing the -5 has further simplified the equation, bringing us closer to isolating the variable x. Now, we have a linear equation without parentheses, which is easier to solve. The next steps will involve isolating the term with x on one side of the equation and then solving for x. This systematic approach, one step at a time, is the key to success in algebra.
After distributing, our equation becomes $5 = -20x + 45$. This equation is now in a simpler form, where we have a linear expression on the right side. Linear equations are generally easier to solve than equations with fractions or parentheses. The goal now is to isolate the term containing x (-20x) on one side of the equation. To do this, we need to eliminate the constant term (45) from the right side. This can be achieved by subtracting 45 from both sides of the equation. Remember, the principle of maintaining equality dictates that any operation performed on one side must also be performed on the other side. This ensures that the balance of the equation is preserved. Subtracting 45 from both sides is a strategic move that brings us closer to isolating x. It's like removing a counterweight from one side of a scale to reveal the true weight on the other side. This step demonstrates the power of algebraic manipulation in simplifying equations and revealing the underlying solution. So, let's proceed with this step and continue our journey towards finding the value of x.
The importance of careful attention to signs cannot be overstated in this step. When distributing and simplifying, pay close attention to the positive and negative signs. A single sign error can throw off the entire solution. For instance, if we had incorrectly distributed the -5 as -20x - 45, it would lead to a completely different and incorrect answer. So, double-checking your work at each step, especially when dealing with negative signs, is a good practice to cultivate. This meticulous approach can save you from making costly mistakes and ensure the accuracy of your solutions. In mathematics, precision is paramount, and even seemingly small errors can have significant consequences. So, take the time to carefully review your work and pay close attention to the details, particularly the signs. This will build confidence in your ability to solve equations accurately and efficiently. With a solid understanding of the distributive property and careful attention to signs, you can confidently tackle a wide range of algebraic problems.
Step 3: Isolate the x Term
To isolate the x term, we subtract 45 from both sides of the equation: $5 - 45 = -20x + 45 - 45$. This simplifies to $-40 = -20x$. By subtracting 45 from both sides, we have effectively removed the constant term from the right side, leaving only the term containing x. This is a crucial step in isolating the variable. It's like peeling away another layer surrounding x, bringing us closer to its value. The principle of equality is once again at play here. Subtracting the same value from both sides ensures that the equation remains balanced and the solution remains valid. This strategic manipulation of the equation is a hallmark of algebraic problem-solving. By carefully choosing the operations to perform, we can gradually simplify the equation and reveal the solution. This step-by-step approach is the key to success in algebra. So, let's continue our journey, armed with this simplified equation, and move towards the final step of solving for x.
The equation -40 = -20x is now in a very simple form. The term containing x (-20x) is isolated on the right side, and a constant term (-40) is on the left side. The next and final step is to isolate x itself. This can be achieved by dividing both sides of the equation by the coefficient of x, which is -20. Remember, the coefficient is the number that multiplies the variable. Dividing both sides by the same non-zero value is another application of the principle of equality. It's like splitting a balanced scale into equal parts – each part remains balanced. This final step will reveal the value of x that satisfies the original equation. So, let's proceed with dividing both sides by -20 and complete our solution journey.
It's worth emphasizing the importance of performing the same operation on both sides of the equation. This is the cornerstone of equation solving. It ensures that the equality is maintained throughout the process. Any deviation from this principle can lead to an incorrect solution. Think of it as a golden rule in algebra – always treat both sides of the equation equally. This principle applies to all algebraic operations, including addition, subtraction, multiplication, division, and even more complex operations like taking the square root. By adhering to this rule, you can confidently manipulate equations and arrive at the correct solutions. So, always double-check that you are performing the same operation on both sides of the equation to avoid errors and ensure the validity of your results. This consistent application of the principle of equality is the foundation of successful equation solving.
Step 4: Solve for x
Finally, to solve for x, we divide both sides of the equation by -20: $ rac{-40}{-20} = rac{-20x}{-20}$. This simplifies to $x = 2$. By dividing both sides by -20, we have successfully isolated x and found its value. This is the culmination of our step-by-step journey through the equation-solving process. It's like reaching the summit of a mountain after a challenging climb – the view is rewarding. The solution x = 2 is the value that, when substituted back into the original equation, makes the equation true. This is the ultimate test of the validity of our solution. We have systematically applied algebraic principles and techniques to unravel the equation and arrive at this answer. This process demonstrates the power of algebra in solving problems and revealing hidden relationships.
Now that we have found the solution, it is crucial to verify it. This step ensures that our solution is correct and that we haven't made any errors along the way. To verify the solution, we substitute x = 2 back into the original equation: $ rac5}{4(2) - 9} = -5$. Simplifying the denominator, we get8 - 9} = -5$. This further simplifies to{-1} = -5$. Finally, we have: $-5 = -5$. Since the equation holds true, our solution x = 2 is correct. This verification step is an essential part of the problem-solving process. It provides assurance that our efforts have been successful and that we have arrived at the correct answer. It also helps to identify any potential errors that may have been made during the solution process. So, always remember to verify your solutions, especially in mathematical problems. This practice will not only enhance your understanding but also build confidence in your problem-solving abilities.
Before declaring our final answer, let's revisit the potential pitfall we identified earlier: ensuring that our solution does not make the original denominator equal to zero. We found that x e rac{9}{4}. Since our solution is x = 2, which is not equal to rac{9}{4}, our solution is valid. This final check is a critical step in solving equations involving fractions. It ensures that our solution is not an extraneous solution, which is a value that satisfies the transformed equation but not the original equation. By checking for extraneous solutions, we can avoid incorrect answers and maintain the integrity of our solution process. This meticulous approach is a hallmark of strong mathematical skills. So, always remember to perform this final check when dealing with equations involving fractions or other expressions that may have restricted values. With this final verification, we can confidently conclude that x = 2 is the correct solution to the equation.
Conclusion
Therefore, the solution to the equation $ rac{5}{4x-9} = -5$ is $x = 2$. We have successfully navigated the equation-solving process, step by step, and arrived at the solution. This journey has highlighted the importance of fundamental algebraic principles, such as the principle of equality, the distributive property, and the order of operations. We have also emphasized the significance of verification and checking for extraneous solutions. By mastering these techniques, you can confidently tackle a wide range of algebraic problems and develop a strong foundation in mathematics. Equation solving is a crucial skill that extends beyond the classroom and into various fields of study and real-life applications. So, continue to practice and hone your skills, and you will be well-equipped to face any algebraic challenge that comes your way. The world of mathematics is vast and fascinating, and the ability to solve equations is a key that unlocks its many mysteries.
