Solving For Two Numbers With A Given Sum And Difference

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This article delves into the fascinating realm of algebra, specifically focusing on solving simultaneous equations to determine two unknown numbers. We'll explore a classic problem: the sum of two numbers is 88, and their difference is 24. By designating the larger number as 'x' and the smaller number as 'y', we'll embark on a journey to unravel these numerical mysteries. This comprehensive guide will not only demonstrate how to formulate simultaneous equations from the given information but also elucidate the step-by-step process of solving them. Whether you're a student grappling with algebra concepts or simply an enthusiast eager to sharpen your problem-solving skills, this article promises to equip you with the knowledge and techniques to tackle similar challenges with confidence.

Problem Statement: Unveiling the Unknowns

Let's restate the problem clearly: Two numbers exist, an elusive larger number and its smaller counterpart. When these numbers unite in a sum, their total reaches 88. Conversely, when the smaller number is subtracted from the larger, the difference stands at 24. Our mission is to unmask these numerical identities, assigning the variable 'x' to the larger number and 'y' to the smaller. To achieve this, we will transition the word problem into a mathematical form. We will formulate simultaneous equations to represent the given relationships and then proceed to solve them. The beauty of algebra lies in its ability to translate real-world scenarios into symbolic representations, allowing us to manipulate and solve for unknown quantities. This process not only helps us find the specific numbers in question but also provides a framework for solving a wide range of problems involving relationships between quantities. So, let's dive in and decode this numerical puzzle together.

(a) Crafting Simultaneous Equations: The Language of Algebra

Our first step is to translate the verbal information into the language of algebra. We are given two crucial pieces of information: the sum of the two numbers and their difference. Remember, 'x' represents the larger number, and 'y' represents the smaller number.

  • The sum of the two numbers is 88: This translates directly into the equation x + y = 88. This equation embodies the additive relationship between the two numbers, stating that their combined value equals 88. It's a fundamental representation of the first piece of information provided in the problem. Understanding how to translate verbal statements into algebraic equations is a cornerstone of mathematical problem-solving.

  • Their difference is 24: This transforms into the equation x - y = 24. This equation captures the subtractive relationship between the numbers, indicating that the larger number ('x') exceeds the smaller number ('y') by 24. It's the second key equation in our system, providing another perspective on the relationship between 'x' and 'y'.

Thus, we have successfully constructed two simultaneous equations:

  1. x + y = 88
  2. x - y = 24

These equations form a system, meaning they are considered together to find a solution that satisfies both conditions simultaneously. Now, we have a mathematical representation of the problem, setting the stage for solving for 'x' and 'y'. The ability to formulate these equations is a crucial skill in algebra, allowing us to tackle a wide array of problems involving relationships between unknown quantities. We'll move on to the next step: solving these equations to unveil the values of 'x' and 'y'.

(b) Solving for the Unknowns: A Journey to the Solution

Now that we have our two simultaneous equations, the next challenge is to solve for the values of 'x' and 'y'. There are several methods to achieve this, but we will employ the elimination method, a technique particularly well-suited for this problem due to the opposing signs of the 'y' terms.

The Elimination Method: A Strategic Approach

The elimination method hinges on the principle of adding or subtracting equations to eliminate one variable, leaving us with a single equation in one unknown. In our case, notice that the 'y' terms have opposite signs (+y and -y). This makes the elimination method an efficient choice.

  1. Add the two equations: By adding the equations (x + y = 88) and (x - y = 24), we strategically eliminate 'y'. (x + y) + (x - y) = 88 + 24 This simplifies to: 2x = 112

  2. Solve for x: Now we have a simple equation with just one variable. Divide both sides of the equation by 2 to isolate 'x'. 2x / 2 = 112 / 2 x = 56

    We have successfully determined the value of 'x', which represents the larger number. We now know that the larger number is 56. But our journey isn't complete yet. We still need to find the value of 'y', the smaller number.

  3. Substitute the value of x: To find 'y', we substitute the value of 'x' (56) into either of the original equations. Let's use the first equation, x + y = 88. 56 + y = 88

  4. Solve for y: Subtract 56 from both sides of the equation to isolate 'y'. 56 + y - 56 = 88 - 56 y = 32

    We have found the value of 'y'! The smaller number is 32.

The Solution Unveiled

We have successfully navigated the algebraic landscape and arrived at the solution. Through the power of simultaneous equations and the strategic elimination method, we have discovered the two numbers:

  • x = 56 (The larger number)
  • y = 32 (The smaller number)

To ensure the accuracy of our solution, it's always a good practice to verify our answers. Let's check if these numbers satisfy the original conditions.

  • Sum: 56 + 32 = 88 (Correct!)
  • Difference: 56 - 32 = 24 (Correct!)

Our solution satisfies both conditions, confirming that our calculations are accurate. We have successfully solved the problem, demonstrating the power and elegance of algebra in unraveling numerical mysteries. The elimination method, as we've seen, is a valuable tool in our problem-solving arsenal, particularly effective when dealing with equations containing variables with opposing signs.

Conclusion: Mastering the Art of Problem Solving

In this exploration, we successfully navigated a classic algebraic problem involving simultaneous equations. We transformed a word problem into mathematical equations, employed the elimination method to strategically solve for the unknowns, and verified our solution to ensure accuracy. The key takeaways from this journey are:

  • Translating words into equations: The ability to represent real-world scenarios using algebraic equations is fundamental to problem-solving.
  • The power of simultaneous equations: These systems allow us to solve for multiple unknowns by considering relationships between them.
  • Strategic method selection: Choosing the appropriate method, such as elimination, can significantly simplify the solution process.
  • Verification is key: Always verify your solution against the original conditions to ensure accuracy.

By mastering these concepts and techniques, you will be well-equipped to tackle a wide range of algebraic problems with confidence. The world of mathematics is filled with puzzles waiting to be solved, and with the right tools and approach, you can unlock their secrets. Remember, practice is paramount. The more you engage with these concepts, the more comfortable and proficient you will become in the art of problem-solving. Embrace the challenge, and let the journey of mathematical discovery continue!