Solve For X And Y: Complex Number Equation

When we're faced with an equation involving complex numbers, like $3 x+8 i=21+4 y i$, it might seem a little intimidating at first. But fear not! Solving for the unknown variables, x and y, in this scenario is quite straightforward once you understand the fundamental principle of equating real and imaginary parts. This principle stems from the very definition of complex number equality. Two complex numbers are considered equal if, and only if, their real components are equal and their imaginary components are equal. This concept is the cornerstone of how we'll unpack this equation and find the unique values for x and y. It's like having two pairs of shoes – you can only say the pairs are identical if the left shoe from one pair matches the left shoe from the other, and the right shoe from one pair matches the right shoe from the other. We can't mix and match; the correspondence must be exact for each part. This is precisely how we approach complex numbers. We'll meticulously separate the real parts from the imaginary parts on both sides of the equation and then set up two simple, independent linear equations. These equations will be much easier to handle, and solving them will reveal the values of x and y that satisfy the original complex number equation. Let's dive in and break down this equation step-by-step, making the process clear and understandable for everyone.

Understanding Complex Number Equality

The core idea behind solving equations like $3 x+8 i=21+4 y i$ lies in a crucial property of complex numbers: equality. For any two complex numbers, say $a + bi$ and $c + di$, to be equal ($a + bi = c + di$), two conditions must be met simultaneously. First, the real parts must be identical, meaning $a = c$. Second, the imaginary parts must also be identical, meaning $b = d$. This is because a complex number has two distinct components: a real part and an imaginary part. When we equate two complex numbers, we are essentially asserting that both of these components match up perfectly. Think of it like comparing two addresses; for them to be the same, the street number, street name, city, state, and zip code must all match. You can't have one address with the same street number but a different city and call it identical. The same logic applies to complex numbers. The real part of the first complex number must equal the real part of the second, and the imaginary part of the first must equal the imaginary part of the second. This principle is fundamental and allows us to transform a single complex equation into a system of two simpler, real-valued equations, which are typically much easier to solve. This systematic approach ensures that we find the values of the variables that make the entire complex number statement true, not just a part of it. By separating and equating these components, we can isolate the unknowns and derive concrete solutions for x and y. This method is robust and widely applicable in various fields of mathematics and engineering where complex numbers play a significant role.

Equating Real and Imaginary Parts

Now, let's apply this principle to our specific equation: $3 x+8 i=21+4 y i$. Our first task is to clearly identify the real and imaginary parts on both sides of the equals sign. On the left side, $3x$ is the real part because it doesn't have an '$i$' attached to it. The term $8i$ is the imaginary part because it is multiplied by the imaginary unit '$i$'. So, for the left side, we have: Real part = $3x$, Imaginary part = $8$.

On the right side of the equation, we have $21 + 4yi$. Here, $21$ is the real part, as it's a standalone number. The term $4yi$ is the imaginary part because it's multiplied by '$i$'. So, for the right side, we have: Real part = $21$, Imaginary part = $4y$.

Having identified these components, we can now use the principle of complex number equality. We set the real parts equal to each other and the imaginary parts equal to each other. This gives us two separate equations:

1. Equating the real parts: $3x = 21$
2. Equating the imaginary parts: $8 = 4y$

These are now two simple linear equations with one variable each. They are much more manageable than the original complex equation. By solving these individually, we can find the values of x and y that satisfy the original problem. This transformation is the key step that simplifies the problem and makes it solvable using basic algebraic techniques. It highlights the power of understanding the underlying mathematical definitions and properties.

Solving for x

We have established that by equating the real parts of our complex equation $3 x+8 i=21+4 y i$, we get the equation: $3x = 21$. This is a straightforward linear equation where our goal is to isolate the variable x. To do this, we need to perform the inverse operation of multiplication, which is division. Currently, x is being multiplied by 3. To get x by itself, we will divide both sides of the equation by 3. It is crucial to perform the same operation on both sides of the equation to maintain the equality. So, we have:

$$\frac{3x}{3} = \frac{21}{3}$$

Simplifying both sides, we get:

$$x = 7$$

Thus, we have found the value of x. This value, $x=7$, is the specific real number that, when substituted into the real part of the original complex equation on the left side ($3x$), will make it equal to the real part on the right side ($21$). This step is vital for ensuring the overall equality of the complex numbers. It's a direct application of basic algebra, demonstrating how the properties of complex numbers allow us to break down complex problems into simpler, solvable parts. The process is methodical: identify the real components, set them equal, and solve the resulting algebraic equation. This ensures that the real-number conditions for equality are met precisely.

Solving for y

Following the same logical progression, we now turn our attention to the imaginary parts of the equation $3 x+8 i=21+4 y i$. By equating the imaginary components, we arrived at the equation: $8 = 4y$. Our objective here is to solve for the variable y. Similar to solving for x, this is a simple linear equation, and we need to isolate y. Currently, y is being multiplied by 4. To solve for y, we will perform the inverse operation, which is division, on both sides of the equation. This ensures that the equality remains valid.

So, we divide both sides by 4:

$$\frac{8}{4} = \frac{4y}{4}$$

Performing the division on both sides yields:

$$2 = y$$

Or, more conventionally written:

$$y = 2$$

This means that the value $y=2$ is the specific real number that, when substituted into the imaginary part of the original complex equation on the right side ($4yi$), will make it equal to the imaginary part on the left side ($8i$, where the imaginary coefficient is 8). This step is just as critical as solving for x because both the real and imaginary parts must match for the complex numbers to be considered equal. We have successfully found both x and y by applying the fundamental rule of equating the real and imaginary parts separately. The process is complete, and we have our solutions.

Verification of the Solution

To ensure that our calculated values for x and y are correct, it's always a good practice to verify the solution by substituting them back into the original complex equation: $3 x+8 i=21+4 y i$. We found that $x=7$ and $y=2$. Let's plug these values in.

Substitute $x=7$ into the left side:

$$3(7) + 8i = 21 + 8i$$

Now, substitute $y=2$ into the right side:

$$21 + 4(2)i = 21 + 8i$$

As you can see, after substituting our values for x and y, the left side of the equation ($21 + 8i$) is exactly equal to the right side of the equation ($21 + 8i$). This confirms that our solutions are correct. The real parts match ($21 = 21$), and the imaginary parts match ($8 = 8$). This verification step provides confidence in our calculations and reinforces the understanding of how complex number equality works. It's a satisfying conclusion to the problem, demonstrating that the method of equating real and imaginary parts is effective and leads to accurate results. This process is vital for solidifying understanding and building confidence in tackling similar problems.

Conclusion

In conclusion, solving for x and y in the complex equation $3 x+8 i=21+4 y i$ hinges on the fundamental principle of complex number equality: real parts must equal real parts, and imaginary parts must equal imaginary parts. By carefully separating these components, we transformed the single complex equation into two independent linear equations: $3x = 21$ for the real parts and $8 = 4y$ for the imaginary parts. Solving the first equation yielded $x = 7$, and solving the second equation gave us $y = 2$. We then verified these solutions by substituting them back into the original equation, confirming that $21 + 8i = 21 + 8i$, which is true. This systematic approach, rooted in understanding the structure of complex numbers, allows us to confidently find the values of unknown variables in such equations. This method is not only applicable to this specific problem but also serves as a foundational technique for more advanced problems involving complex numbers in various mathematical and scientific disciplines. If you're interested in learning more about complex numbers and their properties, exploring resources on complex analysis can provide deeper insights into this fascinating area of mathematics.

For further exploration into the fascinating world of complex numbers and their applications, you might find the resources at Wolfram MathWorld to be incredibly helpful. Their comprehensive explanations and examples can deepen your understanding significantly.