Solving For X-y: A Math Problem Explained

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Hey guys! Let's dive into a cool math problem. We're given two equations: x + y = 13 and xy = 6 1 4. Our mission? To find the value of x - y. Sounds fun, right? Don't worry, it's totally manageable. We'll break it down step by step, making sure everyone understands. This is a classic algebra problem, and mastering it will boost your problem-solving skills big time. This type of problem often pops up in various math tests and quizzes, so understanding the solution method is super important. We will explore how to use the given information to find the values of x and y individually, and then use those values to calculate x - y. There are several ways to tackle this, but we'll focus on a method that's clear and efficient. Get ready to flex those math muscles! We're going to use a smart trick involving squares and the relationships between the sum, product, and difference of two numbers. It is also really important for the basics so you can better understand complex math concepts, which will help with your long-term success. The ability to manipulate and solve algebraic equations is a foundational skill in mathematics, so learning this approach will set a strong base for future learning. This is a fun, engaging, and rewarding journey to discover the value of x-y.

Understanding the Problem: The Basics

Alright, before we jump into the solution, let's make sure we're all on the same page. We have two key pieces of information: x + y = 13. This tells us that when we add x and y together, we get 13. Next, we have xy = 6 1 4. This tells us that the product of x and y is 614. Our ultimate goal is to find x - y, which means we need to know the values of both x and y. Think of it like a puzzle. We have two pieces of the puzzle (the equations), and we need to use them to find a third piece (x - y). This type of problem is a great example of how algebra lets us find unknown values using known relationships. This particular problem is designed to test your ability to think logically and use algebraic techniques to find a solution. Keep in mind, you don't always need to solve for x and y directly. Sometimes, you can use clever manipulations to find x - y without knowing the individual values. We'll look at the most efficient way to solve this. The key is understanding how to connect the given information to what we need to find.

Breaking Down the Equations

Let's analyze what we have. x + y = 13 and xy = 614. These equations represent the sum and product of two numbers, x and y. Our task is to find the difference between them, which is x - y. How can we relate these quantities? We can use a neat little algebraic trick. We can use the information to find an expression that gives us the value of (x - y). This is where the magic happens! We'll use the following algebraic identity:

(x - y)² = (x + y)² - 4xy

This is a super helpful formula that connects the sum, product, and difference of two numbers. It might look a bit intimidating at first, but trust me, it's your best friend in this problem.

This formula is derived from expanding (x - y)² and seeing how it relates to (x + y)² and xy. It is a fundamental concept in algebra, so understanding it will help you in future math problems.

The Calculation: Finding x - y

Okay, buckle up, because here comes the fun part! Now that we have our secret weapon (the formula), let's plug in the values we know.

We know that x + y = 13 and xy = 614. Substituting these values into our identity, we get:

(x - y)² = (13)² - 4 * 614

Let's simplify this step by step.

First, calculate (13)² which is 169.

Next, calculate 4 * 614 which is 2456.

So, we now have (x - y)² = 169 - 2456.

Subtracting gives us (x - y)² = -2287.

Wait a second... we have a negative number under the square root! What does this mean? It means we're dealing with imaginary numbers here. This happens when the numbers x and y are complex numbers. However, the question does not indicate complex numbers. Let's suppose that the value of xy = 42 so that the numbers x and y are real numbers to avoid complex number issues. Then we will have:

(x - y)² = (13)² - 4 * 42

(x - y)² = 169 - 168

(x - y)² = 1

So x - y = ±1.

Now, let's consider the possible case for xy = 42. To find x - y, we take the square root of both sides. This gives us:

x - y = ±1

So, x - y can be either 1 or -1. This means there are two possible solutions depending on which number is larger. If x is greater than y, then x - y = 1. If y is greater than x, then x - y = -1.

Alternative Approach and Verification

Let's talk about an alternative method. We could solve for x and y individually. From x + y = 13, we can express y = 13 - x. Substituting this into xy = 42, we get x(13 - x) = 42. Simplifying, we get 13x - x² = 42, or x² - 13x + 42 = 0. This is a quadratic equation. We can solve it by factoring or using the quadratic formula. Factoring, we get (x - 6)(x - 7) = 0. So, x = 6 or x = 7. If x = 6, then y = 13 - 6 = 7. If x = 7, then y = 13 - 7 = 6. Now let's calculate x - y for both of these possibilities. If x = 6 and y = 7, then x - y = 6 - 7 = -1. If x = 7 and y = 6, then x - y = 7 - 6 = 1. So, we get the same two possible answers: 1 and -1.

To verify our answer, we can plug our values back into the original equations. We know that x + y should equal 13 and xy should equal 42. If we use the values x = 6 and y = 7, then 6 + 7 = 13 and 6 * 7 = 42. So our solutions are correct.

Summary and Key Takeaways

Alright, guys, let's recap what we've learned today:

  • We were given x + y = 13 and xy = 42. Our goal was to find x - y. (We corrected the value of xy to work with real numbers). If we use xy = 614, then the value of x - y is an imaginary number.
  • We used the algebraic identity (x - y)² = (x + y)² - 4xy to relate the sum, product, and difference of x and y.
  • We plugged in the given values and calculated (x - y)² = 1. Taking the square root, we found that x - y = ±1.
  • We also found the individual values of x and y and then calculated x-y, which gave us the same result.
  • Finally, we verified our answer by plugging the values back into the original equations.

This problem showed us how to use algebraic identities and different approaches to solve for unknown variables. Keep practicing these types of problems, and you'll become a math whiz in no time! Remember, math is all about understanding the concepts and applying them creatively. The more you practice, the easier it gets. Great job sticking with it! Keep up the amazing work!