Prove X Is Rational: Solving A/b + X = C/d

Hey guys! Today, we're diving into a cool math problem that involves fractions and rational numbers. We're given an equation where a, b, c, and d are all nonzero integers. The equation states that if we add the fraction a/b to some number x, the result is the fraction c/d. Our mission, should we choose to accept it, is to figure out how to prove that x must be a rational number. Buckle up, because we're about to embark on a mathematical adventure!

Understanding the Problem

Before we jump into the proof, let's make sure we're all on the same page about what the problem is asking. We've got this equation: a/b + x = c/d. Remember, a rational number is any number that can be expressed as a fraction where both the numerator and the denominator are integers (and the denominator isn't zero, of course). The question is essentially asking us to show that the number x in this equation can always be written as a fraction of integers, given that a/b and c/d are also fractions of integers. To really grasp this, let’s break down each component. We know a, b, c, and d are nonzero integers. This is crucial because it sets the stage for dealing with rational numbers. When we say a/b is added to x, resulting in c/d, we’re talking about basic arithmetic operations, but with a twist of abstract algebra. The challenge is to manipulate this equation to isolate x and then demonstrate that the result is indeed a rational number. Think of it as a puzzle where the pieces are integers and fractions, and our goal is to rearrange them to reveal the nature of x. Understanding this setup deeply will help us appreciate the elegance of the solution and why each step is mathematically sound. It’s not just about finding the answer; it’s about understanding the ‘why’ behind it. So, with this solid foundation, we can confidently move forward to explore the proof and the mathematical principles that make it work.

Isolating x: The Key to the Proof

So, how do we even begin to show that x is rational? The first step, and a super important one, is to isolate x on one side of the equation. We want to get x = something, and that something needs to be a fraction that clearly shows it's a rational number. Remember our equation: a/b + x = c/d. To get x by itself, we need to get rid of the a/b term on the left side. We can do this by subtracting a/b from both sides of the equation. This is a fundamental algebraic move – what we do to one side, we must do to the other to keep the equation balanced. So, we subtract a/b from both sides, and we get: x = c/d - a/b. Now, we're getting somewhere! x is isolated, and we have an expression on the right side that involves two fractions. But, it doesn't immediately scream “rational number” just yet. We need to combine these fractions into a single fraction to really see if x fits the definition of a rational number. This step is crucial because it transforms the expression into a form where the numerator and denominator are clearly visible, making it easier to verify if they are integers. So, hang tight, because the next step is where we'll do some fraction magic to combine those terms and reveal the true nature of x.

Combining Fractions: Finding a Common Denominator

Okay, we've got x = c/d - a/b. Now, to combine these two fractions, we need to find a common denominator. This is like finding a common language for the fractions so we can add or subtract them. The easiest way to find a common denominator is to multiply the two denominators together. So, in our case, the common denominator is b * d, or simply bd. Now, we need to rewrite each fraction with this new denominator. To do that, we multiply the numerator and denominator of the first fraction (c/d) by b, and we multiply the numerator and denominator of the second fraction (a/b) by d. This gives us: c/d = (c * b) / (d * b) = bc/bd and a/b = (a * d) / (b * d) = ad/bd. See what we did there? We didn't change the value of the fractions; we just rewrote them in an equivalent form with the common denominator bd. Now we can substitute these back into our equation for x: x = bc/bd - ad/bd. This is super exciting because we're almost there! We have two fractions with the same denominator, which means we can finally combine them into a single fraction. This step is pivotal in revealing whether x is rational, because it directly leads us to expressing x in the form of one integer divided by another. So, let's take the final leap and combine these fractions to see what we get.

The Grand Finale: Proving x is Rational

Alright, we've reached the final step! We've got x = bc/bd - ad/bd. Since these fractions have the same denominator, we can simply subtract the numerators: x = (bc - ad) / bd. Boom! Look at that! We've expressed x as a single fraction. But is it a rational number? To be rational, the numerator and the denominator both need to be integers. Let's check: We know that a, b, c, and d are all nonzero integers. So, bc is the product of two integers, which is also an integer. Similarly, ad is the product of two integers, making it an integer as well. When we subtract two integers (bc and ad), the result (bc - ad) is also an integer. Now, let's look at the denominator: bd. This is the product of two integers (b and d), so it's definitely an integer. And remember, b and d are nonzero, so bd is also nonzero, which is exactly what we need for the denominator of a rational number. So, we've shown that the numerator (bc - ad) is an integer, and the denominator (bd) is a nonzero integer. That means x = (bc - ad) / bd is a fraction of two integers, which perfectly fits the definition of a rational number! We did it! We've successfully proven that x must be a rational number. This whole process shows how we can use basic algebraic manipulations and the definition of rational numbers to solve problems. It's like a mathematical detective story, where we start with clues and follow them step-by-step to uncover the truth. And in this case, the truth is that x, in this equation, is always a rational number. Isn't math cool?

Conclusion: The Beauty of Rational Numbers

So, guys, we've journeyed through this equation, a/b + x = c/d, and we've successfully shown that x must be a rational number. We started by understanding the problem, then we isolated x, found a common denominator, combined the fractions, and finally, we proved that the resulting expression for x fits the very definition of a rational number. This problem highlights the beauty and consistency of mathematics. It shows how we can take basic concepts, like integers and fractions, and use them to build logical arguments and prove interesting results. The key takeaway here is not just the answer, but the process. By understanding how to manipulate equations, combine fractions, and apply definitions, we can tackle a wide range of mathematical challenges. And that's what makes math so powerful and so fascinating. Whether you're a student grappling with algebra or just a curious mind exploring the world of numbers, remember that math is a journey of discovery. Each problem is a puzzle waiting to be solved, and with a little bit of logical thinking and a dash of creativity, you can unlock its secrets. So, keep exploring, keep questioning, and keep enjoying the beauty of mathematics! And remember, every problem, no matter how daunting it may seem at first, is just an opportunity to learn something new and expand your mathematical horizons. Keep up the awesome work, and who knows what mathematical wonders you'll uncover next!