Solving For 125x² + Y² Given 5x + Y = 13 And Xy = 6

by BRAINLY IN FTUNILA 52 views
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Introduction

In this article, we will delve into a fascinating algebraic problem where we aim to determine the value of the expression 125x² + y² given two equations: 5x + y = 13 and xy = 6. This type of problem often appears in mathematical competitions and serves as an excellent exercise in applying algebraic manipulation and problem-solving skills. The beauty of this problem lies in its ability to be solved through multiple approaches, allowing us to explore different algebraic techniques. We will embark on a step-by-step journey, starting with a clear understanding of the problem statement, followed by a strategic selection of methods, and culminating in a detailed solution. Our focus will be not only on arriving at the correct answer but also on understanding the underlying principles and techniques that make the solution possible. Along the way, we will highlight the importance of algebraic identities, such as (a + b)² = a² + 2ab + b², and how they can be effectively utilized to simplify complex expressions. The problem-solving process will involve isolating variables, squaring equations, and strategically combining information from the given equations to arrive at the desired expression. Moreover, we will discuss common pitfalls and errors to avoid when tackling such problems, ensuring a thorough and accurate approach. By the end of this exploration, you will have gained a deeper appreciation for algebraic problem-solving and developed valuable skills applicable to a wide range of mathematical challenges. So, let’s dive in and unravel the solution to this intriguing problem, equipping ourselves with the knowledge and confidence to tackle similar challenges in the future.

Understanding the Problem

Before we begin to solve the problem, it's crucial to clearly understand what we are given and what we need to find. We are given two equations:

  1. 5x + y = 13
  2. xy = 6

Our goal is to find the value of the expression 125x² + y². This expression seems quite different from the given equations, so we'll need to use some algebraic manipulation to bridge the gap. The key is to identify relationships between the given equations and the target expression. We need to strategically utilize the given information to construct terms that resemble parts of 125x² + y². This might involve squaring one of the equations, multiplying equations by appropriate constants, or using algebraic identities to rewrite expressions in a more convenient form. The challenge lies in finding the right combination of steps that will lead us to the desired value. We must also be mindful of potential pitfalls, such as introducing extraneous solutions or making algebraic errors. Therefore, a systematic and careful approach is essential to ensure accuracy and efficiency. Ultimately, our understanding of the problem's requirements will guide us in selecting the most effective solution strategy and avoiding unnecessary complications. By breaking down the problem into smaller, more manageable parts, we can gain clarity and confidence in our ability to tackle this algebraic puzzle. The initial step of comprehension is paramount, as it sets the stage for the entire solution process, ensuring that we are focused on the correct objective and equipped with the necessary tools to achieve it.

Strategy and Approach

To find the value of 125x² + y², we need to relate it to the given equations. A common technique in such problems is to square the first equation (5x + y = 13) because squaring will introduce squared terms. The strategic approach here involves a series of algebraic manipulations designed to transform the given equations into an expression that closely resembles the target, 125x² + y². Initially, we recognize that squaring the first equation will generate terms of the form (5x)² and y², which are components of our target expression. However, squaring will also introduce a cross-term, 2(5x)(y), which we will need to address. This is where the second equation, xy = 6, comes into play. By substituting the value of xy from the second equation into the expanded form of the squared first equation, we can eliminate the cross-term and create an equation involving only squared terms and constants. The next phase of our strategy involves isolating the desired expression, 125x² + y². We observe that the squared term (5x)² from the expanded equation is 25x², not 125x², so we will need to multiply the entire equation by a suitable constant to achieve the desired coefficient of 125. Once we have an equation with 125x² and y² terms, we can perform the necessary algebraic operations to isolate the expression and determine its value. This might involve adding or subtracting constants from both sides of the equation. Throughout the process, we must remain vigilant to avoid common errors such as incorrect squaring, misapplication of algebraic identities, or arithmetic mistakes. A systematic approach, with each step carefully justified, is crucial to ensure accuracy and clarity. By meticulously executing this strategy, we will navigate the problem towards a successful resolution, ultimately finding the value of 125x² + y².

Step-by-Step Solution

  1. Square the first equation: (5x + y)² = 13²

    Expanding this, we get: 25x² + 10xy + y² = 169.

    Squaring the equation 5x + y = 13 is a pivotal step in our solution strategy. This manipulation allows us to introduce the squared terms, 25x² and y², which are essential components of the expression we aim to evaluate, 125x² + y². The expansion of (5x + y)² using the algebraic identity (a + b)² = a² + 2ab + b² yields 25x² + 10xy + y². The resulting equation, 25x² + 10xy + y² = 169, now includes terms that are directly related to our target expression. However, we also encounter an additional term, 10xy, which needs to be addressed. This term represents the cross-product of 5x and y and is crucial in linking the first equation to the second equation, xy = 6. By incorporating the information from the second equation, we can eliminate this cross-term and simplify the equation further. The significance of this step lies in its ability to transform the original equation into a form that is more amenable to solving for our target expression. It demonstrates the power of algebraic manipulation in reshaping equations to reveal hidden relationships and facilitate the solution process. By carefully expanding the squared equation, we have laid the groundwork for subsequent steps, which will involve substituting the value of xy and isolating the desired expression. This methodical approach ensures that we are progressing towards the solution in a structured and logical manner.

  2. Substitute xy = 6 into the expanded equation:

    25x² + 10(6) + y² = 169

    25x² + 60 + y² = 169

    Substituting the value of xy from the second equation into the expanded form of the squared first equation is a critical step in simplifying the expression and moving closer to our goal. We know that xy = 6, and the expanded equation from the previous step is 25x² + 10xy + y² = 169. By replacing xy with its numerical value, we effectively eliminate the mixed term involving both x and y, leaving us with an equation that contains only squared terms and constants. This substitution transforms the equation into 25x² + 10(6) + y² = 169, which simplifies to 25x² + 60 + y² = 169. The result is a significant advancement because it now directly relates the squared terms, 25x² and y², to a constant value. This is a key step in isolating the expression we are interested in, 125x² + y². The strategic substitution of known values is a fundamental technique in algebraic problem-solving. It allows us to reduce the complexity of equations and uncover hidden relationships between variables. In this case, the substitution of xy = 6 bridges the gap between the two given equations and enables us to express the problem in terms of squared terms, bringing us one step closer to finding the value of 125x² + y². By carefully performing this substitution, we have streamlined the equation and set the stage for further manipulations that will lead us to the final solution. This methodical approach underscores the importance of leveraging all available information to simplify and solve algebraic problems.

  3. Rearrange the equation:

    25x² + y² = 169 - 60

    25x² + y² = 109

    Rearranging the equation 25x² + 60 + y² = 169 is a crucial step in isolating the terms that are relevant to our target expression, 125x² + y². By subtracting 60 from both sides of the equation, we isolate the sum of the squared terms, 25x² + y², on the left-hand side. This manipulation brings us closer to our objective by grouping the terms that we need to evaluate. The resulting equation, 25x² + y² = 109, provides a direct relationship between the squared terms and a constant value. This simplified equation is a significant milestone in our solution process. It represents a condensed form of the information we have gathered from the initial equations and the subsequent algebraic manipulations. Now, we have a clear and concise equation that focuses on the key components of our target expression. The importance of this step lies in its ability to distill the problem into its essential elements. By isolating the relevant terms, we can now concentrate our efforts on transforming the equation further to match the form of 125x² + y². The strategic rearrangement of equations is a fundamental technique in algebra, allowing us to manipulate expressions to highlight the relationships between variables and constants. In this case, the rearrangement has brought us closer to our final goal by isolating the squared terms and setting the stage for the next phase of the solution process. This methodical approach ensures that we are making steady progress towards finding the value of 125x² + y².

  4. Multiply both sides by 5:

    5(25x² + y²) = 5(109)

    125x² + 5y² = 545

    Multiplying both sides of the equation 25x² + y² = 109 by 5 is a strategic maneuver aimed at obtaining the desired coefficient of 125 for the x² term. Our target expression is 125x² + y², and we currently have 25x² + y² = 109. By multiplying the entire equation by 5, we transform the 25x² term into 125x², bringing us significantly closer to our goal. The resulting equation is 125x² + 5y² = 545. While we have successfully obtained the 125x² term, we have also introduced a 5y² term, which is different from the y² term in our target expression. This means that we are not quite at the final answer yet and need to make an adjustment to account for this additional 4y². The significance of this step lies in its targeted approach to modifying the equation. We specifically chose to multiply by 5 to achieve the desired coefficient for the x² term, demonstrating a clear understanding of the problem's requirements. Although we now have an extra term to deal with, we have made significant progress towards our final solution. The next step will likely involve finding a way to eliminate or adjust for the 4y² term, which will require further algebraic manipulation. The methodical approach of multiplying by a specific constant to achieve a particular goal is a common technique in algebra. It allows us to strategically transform equations and bring them closer to the desired form. By carefully considering the coefficients and terms involved, we can make informed decisions about how to manipulate the equation effectively.

  5. We want 125x² + y², but we have 125x² + 5y². We need to subtract 4y² from both sides. However, we don't know the value of y² yet. Let's go back to the equation 25x² + y² = 109 and isolate y²:

    y² = 109 - 25x²

    Isolating y² in the equation 25x² + y² = 109 is a crucial step in determining the value of 125x² + y². We've realized that we need to subtract 4y² from the equation 125x² + 5y² = 545 to arrive at our target expression. However, we don't yet know the value of y². By rearranging the equation 25x² + y² = 109, we can express y² in terms of x². This allows us to substitute the expression for y² into our subsequent calculations, effectively eliminating y² as an independent variable and allowing us to work with x² instead. Subtracting 25x² from both sides of the equation yields y² = 109 - 25x². This equation provides a direct relationship between y² and x², which is a valuable piece of information. The significance of this step lies in its strategic approach to dealing with the unknown value of y². Instead of trying to find the numerical value of y² directly, we have chosen to express it in terms of x², which we already have information about through the given equations. This technique of expressing one variable in terms of another is a common and powerful tool in algebraic problem-solving. It allows us to simplify equations and reduce the number of unknowns, making the problem more manageable. By carefully isolating y², we have set the stage for the next phase of the solution, which will likely involve substituting this expression into our previous equation and simplifying further. This methodical approach ensures that we are making steady progress towards finding the value of 125x² + y².

  6. Now, we need to find the value of x². Go back to 5x + y = 13 and xy = 6. Solve for y in the first equation: y = 13 - 5x.

    Substitute this into the second equation: x(13 - 5x) = 6

    13x - 5x² = 6

    5x² - 13x + 6 = 0

    Finding the value of x² is a critical step in our solution process. We've expressed y² in terms of x², but to proceed further, we need to determine the value of x² itself. To do this, we revisit the original equations, 5x + y = 13 and xy = 6. The strategy here is to eliminate one of the variables and obtain an equation in terms of the other variable. We choose to solve for y in the first equation, yielding y = 13 - 5x. This allows us to substitute this expression for y into the second equation, xy = 6. Substituting y = 13 - 5x into xy = 6 gives us x(13 - 5x) = 6. Expanding this equation results in 13x - 5x² = 6. Rearranging the terms, we obtain a quadratic equation in x: 5x² - 13x + 6 = 0. This quadratic equation is a significant breakthrough because it allows us to solve for x, and consequently, find the value of x². The significance of this step lies in its ability to leverage the given equations to create a solvable equation in a single variable. By eliminating y, we have transformed the system of equations into a more manageable form. The quadratic equation 5x² - 13x + 6 = 0 can be solved using various methods, such as factoring, completing the square, or using the quadratic formula. Once we find the roots of this equation, we can easily determine the value of x² by squaring the solutions. This methodical approach of eliminating variables and solving for the remaining variable is a fundamental technique in algebra. It allows us to unravel complex systems of equations and find the values of the unknowns. By carefully deriving and setting up the quadratic equation, we have set the stage for the next phase of the solution, which will involve solving for x and subsequently finding x².

  7. Solve the quadratic equation: 5x² - 13x + 6 = 0

    This can be factored as (5x - 3)(x - 2) = 0

    So, x = 3/5 or x = 2

    Solving the quadratic equation 5x² - 13x + 6 = 0 is a pivotal step in our quest to find the value of 125x² + y². We derived this equation by eliminating the variable y from the original system of equations and expressing the relationship in terms of x alone. Now, we need to find the roots of this quadratic equation, which will give us the possible values of x. The quadratic equation can be solved using various methods, such as factoring, completing the square, or using the quadratic formula. In this case, the equation is readily factorable. Factoring the quadratic equation 5x² - 13x + 6 = 0, we obtain (5x - 3)(x - 2) = 0. This factored form allows us to easily identify the roots of the equation. Setting each factor equal to zero, we get 5x - 3 = 0 or x - 2 = 0. Solving these linear equations for x, we find two possible values: x = 3/5 or x = 2. These two values of x represent the solutions to the quadratic equation and are crucial for determining the value of x². The significance of this step lies in its ability to provide us with concrete values for x. These values will enable us to calculate x², which is a key component in our target expression, 125x² + y². By carefully factoring the quadratic equation and solving for its roots, we have made significant progress towards our final goal. The next step will involve calculating x² for each value of x and then using these values to find y² and ultimately, the value of 125x² + y². This methodical approach of solving for the variables one step at a time ensures that we are progressing towards the solution in a structured and logical manner.

  8. Calculate x² for both values of x:

    If x = 3/5, then x² = (3/5)² = 9/25

    If x = 2, then x² = 2² = 4

    Calculating x² for both values of x is a straightforward but essential step in our solution process. We have found two possible values for x, namely x = 3/5 and x = 2. To proceed further, we need to determine the corresponding values of x², as x² appears in the expressions we are working with. For x = 3/5, squaring both sides gives us x² = (3/5)² = 9/25. This is the value of x² when x is 3/5. Similarly, for x = 2, squaring both sides gives us x² = 2² = 4. This is the value of x² when x is 2. These two values of x², 9/25 and 4, are crucial for calculating the corresponding values of y² and ultimately, the value of our target expression, 125x² + y². The significance of this step lies in its direct contribution to the evaluation of our target expression. By calculating x² for each possible value of x, we are providing the necessary building blocks for the subsequent calculations. The process of squaring a number is a fundamental arithmetic operation, and in this context, it allows us to transform the values of x into the form required for our algebraic manipulations. This methodical approach of calculating the necessary components step by step ensures that we are progressing towards the solution in a clear and organized manner. By carefully calculating x² for both values of x, we have laid the groundwork for the next phase of the solution, which will involve finding the corresponding values of y² and then evaluating 125x² + y².

  9. Substitute the values of x² into y² = 109 - 25x²:

    If x² = 9/25, then y² = 109 - 25(9/25) = 109 - 9 = 100

    If x² = 4, then y² = 109 - 25(4) = 109 - 100 = 9

    Substituting the values of x² into the equation y² = 109 - 25x² is a critical step in determining the corresponding values of y². We derived this equation earlier by isolating y² in terms of x². Now that we have two possible values for x², namely 9/25 and 4, we can use this equation to find the corresponding values of y². For x² = 9/25, substituting this into the equation gives us y² = 109 - 25(9/25) = 109 - 9 = 100. This means that when x² is 9/25, y² is 100. For x² = 4, substituting this into the equation gives us y² = 109 - 25(4) = 109 - 100 = 9. This means that when x² is 4, y² is 9. These two pairs of values for x² and y² are crucial for evaluating our target expression, 125x² + y². The significance of this step lies in its ability to provide us with the necessary values to calculate the final answer. By carefully substituting the values of x² into the equation and simplifying, we have found the corresponding values of y². This process demonstrates the power of algebraic manipulation in relating variables and finding their values. The methodical approach of substituting known values into equations to find unknowns is a fundamental technique in algebra. It allows us to solve for variables step by step and build towards the final solution. By carefully calculating y² for both values of x², we have laid the groundwork for the final phase of the solution, which will involve substituting these values into the expression 125x² + y².

  10. Calculate 125x² + y² for both pairs of values:

    If x² = 9/25 and y² = 100, then 125x² + y² = 125(9/25) + 100 = 45 + 100 = 145

    If x² = 4 and y² = 9, then 125x² + y² = 125(4) + 9 = 500 + 9 = 509

    Calculating 125x² + y² for both pairs of values is the final step in our solution process. We have found two possible pairs of values for x² and y²: (9/25, 100) and (4, 9). Now, we substitute each pair into the expression 125x² + y² to find the corresponding values of the expression. For x² = 9/25 and y² = 100, substituting these values into the expression gives us 125x² + y² = 125(9/25) + 100 = 45 + 100 = 145. This is the value of the expression when x² is 9/25 and y² is 100. For x² = 4 and y² = 9, substituting these values into the expression gives us 125x² + y² = 125(4) + 9 = 500 + 9 = 509. This is the value of the expression when x² is 4 and y² is 9. Therefore, the possible values of 125x² + y² are 145 and 509. The significance of this step lies in its culmination of all the previous steps into a final answer. By carefully substituting the calculated values of x² and y² into the target expression, we have arrived at the possible solutions. This process demonstrates the power of algebraic manipulation and problem-solving techniques in unraveling complex expressions. The methodical approach of breaking down the problem into smaller steps and solving each step systematically has led us to a successful conclusion. By carefully calculating the value of 125x² + y² for both pairs of values, we have completed the solution and answered the original question.

Final Answer

Therefore, the possible values of 125x² + y² are 145 and 509.

Conclusion

In this comprehensive exploration, we successfully determined the possible values of the expression 125x² + y² given the equations 5x + y = 13 and xy = 6. Our journey involved a strategic combination of algebraic techniques, starting with squaring the first equation to introduce squared terms and then utilizing the second equation to eliminate cross-terms. We encountered a quadratic equation, which we skillfully solved to find the possible values of x. From there, we meticulously calculated x² and y² for each solution, ultimately leading us to the two possible values for 125x² + y²: 145 and 509. The key to success in this problem lies in a systematic approach, careful algebraic manipulation, and a clear understanding of the relationships between the given equations and the target expression. We demonstrated the power of algebraic identities, such as (a + b)² = a² + 2ab + b², and the importance of strategic substitution in simplifying complex expressions. Moreover, we highlighted the significance of solving quadratic equations as a crucial skill in algebraic problem-solving. This problem serves as an excellent example of how multiple algebraic concepts can be integrated to arrive at a solution. It also underscores the importance of perseverance and attention to detail in mathematical problem-solving. By working through this problem step by step, we have not only found the answer but also reinforced our understanding of fundamental algebraic principles. The skills and techniques we have employed here are applicable to a wide range of mathematical challenges, making this exploration a valuable learning experience. As we conclude this journey, we encourage you to apply these techniques to similar problems and continue to hone your algebraic problem-solving skills. Remember, the beauty of mathematics lies in its ability to be both challenging and rewarding, and with practice and dedication, you can master even the most complex problems.