Solving Equations: Real & Imaginary Solutions Explained

Hey there, math enthusiasts! Today, we're diving deep into the world of equations, specifically focusing on how to find both real and imaginary solutions. Don't worry, it's not as scary as it sounds! We'll break down the equation $\frac{9x}{10} - 2 = 3 - \frac{19x}{20}$ step-by-step, making sure you grasp every concept along the way. Whether you're a seasoned algebra pro or just starting out, this guide will provide you with the knowledge and confidence to tackle similar problems. So, grab your pencils and let's get started!

Understanding the Basics: Real vs. Imaginary Numbers

Before we jump into the equation, let's quickly review the types of numbers we'll encounter. This is super important because it sets the foundation for understanding our solutions.

Real Numbers: These are the numbers you're probably most familiar with. They include all rational numbers (like fractions and decimals that terminate or repeat, e.g., $\frac{1}{2}$, 0.75, 2.333...) and irrational numbers (like $\pi$ and $\sqrt{2}$, which have non-repeating, non-terminating decimal expansions). In simple terms, real numbers can be plotted on a number line. They represent quantities we can usually measure in the real world. Think of measuring the length of a table, the temperature outside, or the amount of money in your bank account – all these are examples of real numbers.

Imaginary Numbers: Now, here's where things get interesting! Imaginary numbers are based on the imaginary unit, denoted by i. The imaginary unit i is defined as the square root of -1 ($\sqrt{-1}$). Since you can't take the square root of a negative number in the realm of real numbers, imaginary numbers were created to solve such equations. They might seem abstract, but they're incredibly useful in various fields like electrical engineering, quantum mechanics, and signal processing. An imaginary number is typically written in the form bi, where b is a real number. For example, 2i, -5i, and $\pi*i$ are all imaginary numbers. Imaginary numbers don't exist on the traditional number line; they require a two-dimensional plane, where the horizontal axis represents real numbers, and the vertical axis represents imaginary numbers.

Complex Numbers: These are the granddaddies of the number world, combining both real and imaginary parts. A complex number is written in the form a + bi, where a is the real part, and bi is the imaginary part. For instance, 3 + 2i is a complex number where 3 is the real part and 2i is the imaginary part. The set of complex numbers encompasses all real numbers (where the imaginary part is 0) and all imaginary numbers (where the real part is 0). Complex numbers are essential in solving equations that don't have real solutions, such as quadratic equations with negative discriminants. This is why we're focusing on both real and imaginary solutions – complex numbers provide a complete solution set.

Solving the Equation: Step-by-Step

Alright, let's get to the main event! We're going to solve the equation $\frac{9x}{10} - 2 = 3 - \frac{19x}{20}$. Here's a breakdown of how to find the solutions:

Step 1: Eliminate Fractions. The first thing we want to do is get rid of those pesky fractions. To do this, we'll find the least common multiple (LCM) of the denominators, which are 10 and 20. The LCM of 10 and 20 is 20. Now, multiply every term in the equation by 20:

20 * (\frac{9x}{10}) - 20 * 2 = 20 * 3 - 20 * (\frac{19x}{20})

This simplifies to:

18x - 40 = 60 - 19x

Step 2: Combine Like Terms. Next, we want to get all the x terms on one side of the equation and the constant terms on the other side. Let's add 19x to both sides:

18x + 19x - 40 = 60 - 19x + 19x

This simplifies to:

37x - 40 = 60

Now, add 40 to both sides:

37x - 40 + 40 = 60 + 40

This gives us:

37x = 100

Step 3: Isolate x. To solve for x, we need to isolate it. Divide both sides of the equation by 37:

\frac{37x}{37} = \frac{100}{37}

This leaves us with:

x = \frac{100}{37}

So, the solution to the equation is x = $\frac{100}{37}$. This is a real number, as it's a rational number (a fraction). There are no imaginary components in this solution, so we can be confident that our solution is indeed real. If we had encountered a situation where we needed to take the square root of a negative number, that's when we would see imaginary solutions appear.

Verifying the Solution

It's always a good idea to check your answer. Let's plug x = $\frac{100}{37}$ back into the original equation to make sure it holds true.

\frac{9 * (\frac{100}{37})}{10} - 2 = 3 - \frac{19 * (\frac{100}{37})}{20}

Simplifying this, we get:

\frac{900}{370} - 2 = 3 - \frac{1900}{740}

Further simplification:

\frac{90}{37} - 2 = 3 - \frac{190}{74}

Which equals:

\frac{90}{37} - \frac{74}{37} = 3 - \frac{95}{37}

Then:

\frac{16}{37} = \frac{111}{37} - \frac{95}{37}

Finally:

\frac{16}{37} = \frac{16}{37}

Since both sides are equal, our solution x = $\frac{100}{37}$ is correct! This confirmation step is crucial because it helps to identify any potential arithmetic mistakes and ensures you have the accurate answer.

Conclusion: Real Solutions are the Key

So there you have it, guys! We've successfully solved the equation $\frac{9x}{10} - 2 = 3 - \frac{19x}{20}$ and found that the solution is a real number, x = $\frac{100}{37}$. This example highlights the fundamental process of solving equations. The skills gained from solving this equation can easily be applied to more complex mathematical problems. Keep practicing and remember the basics! Understanding the difference between real and imaginary numbers is essential for advanced mathematics, especially when dealing with quadratic equations, complex numbers, and more. Keep exploring and happy solving!

If you enjoyed this explanation, be sure to check out our other guides on algebra and calculus! We regularly post new content to enhance your mathematical journey, so don't miss out. Learning math can be a fun adventure if you approach it step by step and enjoy the problem-solving process. Let us know what mathematical topics you would like us to cover next, and don't forget to like and share this tutorial with your friends and classmates! See you in the next lesson! Feel free to ask any questions in the comments below. We're here to help you succeed!