Solving For X And Y In The Equation (x+yi)+(4+9i)=9-4i
At times, in the realm of mathematics, we encounter equations that seem perplexing at first glance. Complex equations involving imaginary numbers can appear daunting, but with a systematic approach, they can be solved elegantly. This article delves into the process of solving a complex equation, guiding you through the steps to find the values of the variables that satisfy the equation. Let's embark on this mathematical journey and unravel the mysteries of complex numbers.
Before we plunge into solving the equation, it's crucial to have a firm grasp on the concept of complex numbers. Complex numbers, denoted by the form a + bi, where a and b are real numbers, and i represents the imaginary unit. The imaginary unit, i, is defined as the square root of -1, i.e., i^2 = -1. The real part of the complex number is a, and the imaginary part is b. Complex numbers extend the realm of real numbers by incorporating imaginary components, opening up new avenues in mathematical exploration.
Complex numbers are composed of two parts a real part and an imaginary part. The real part is a standard real number, while the imaginary part is a real number multiplied by the imaginary unit i, where i is defined as the square root of -1. Understanding complex numbers is essential for solving equations like the one we are about to tackle. When dealing with complex equations, we treat the real and imaginary parts separately. This means that for two complex numbers to be equal, their real parts must be equal, and their imaginary parts must be equal. This principle is fundamental to solving for unknowns in complex equations.
Let's consider the equation at hand:
(x + yi) + (4 + 9i) = 9 - 4i
This equation involves complex numbers on both sides of the equals sign. Our mission is to determine the values of x and y that make this equation true. To accomplish this, we'll need to leverage the properties of complex numbers and algebraic manipulation. In this equation, we have two complex numbers being added on the left side, resulting in another complex number on the right side. The key to solving this equation lies in understanding that we can equate the real parts and the imaginary parts separately. This is because complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal. By separating the real and imaginary parts, we can form two separate equations, each with one unknown, which are much easier to solve. This approach allows us to break down a complex problem into simpler, more manageable steps, making the solution more accessible.
To solve for x and y, we'll follow these steps:
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Group the real and imaginary terms:
(x + 4) + (y + 9)i = 9 - 4i
We begin by grouping the real parts and the imaginary parts on the left side of the equation. This involves combining the real numbers x and 4, and the imaginary terms yi and 9i. By doing this, we create a clearer picture of the equation's structure, setting the stage for separating the real and imaginary components. This step is crucial because it allows us to apply the principle that two complex numbers are equal if and only if their real and imaginary parts are equal. By organizing the equation in this manner, we make it easier to identify and equate the corresponding parts, which is the key to solving for the unknowns x and y. This methodical approach is a cornerstone of solving complex equations effectively.
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Equate the real parts:
x + 4 = 9
Now, we equate the real parts from both sides of the equation. This means setting the real part on the left side, which is x + 4, equal to the real part on the right side, which is 9. This step is based on the fundamental property of complex numbers that states for two complex numbers to be equal, their real parts must be equal. By equating the real parts, we create a simple algebraic equation with one unknown, x. This equation is much easier to solve than the original complex equation, allowing us to isolate x and find its value. This step is a critical part of the solution process, as it allows us to determine one of the unknowns and move closer to the complete solution.
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Solve for x:
x = 9 - 4
x = 5
To solve for x, we subtract 4 from both sides of the equation. This isolates x on the left side, giving us the value of x. The result, x = 5, is a crucial piece of the puzzle. This step is a direct application of basic algebraic principles, demonstrating how fundamental math skills are essential even when dealing with complex numbers. By isolating and solving for x, we have determined one of the two unknowns, bringing us closer to fully solving the original equation. This clear and concise algebraic manipulation is a hallmark of effective problem-solving in mathematics.
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Equate the imaginary parts:
y + 9 = -4
Next, we equate the imaginary parts from both sides of the equation. This involves setting the coefficient of i on the left side, which is y + 9, equal to the coefficient of i on the right side, which is -4. This step mirrors the process we used for the real parts, applying the same principle that the imaginary parts of two equal complex numbers must also be equal. By equating the imaginary parts, we create another simple algebraic equation with one unknown, y. This equation allows us to isolate and solve for y, just as we did for x, bringing us closer to the complete solution of the complex equation. This step is vital for finding the value of y and completing the solution.
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Solve for y:
y = -4 - 9
y = -13
To solve for y, we subtract 9 from both sides of the equation. This isolates y on the left side, giving us the value of y. The result, y = -13, completes the solution. This step is a straightforward algebraic manipulation, similar to what we did for x, demonstrating the consistency of the solution process. By finding the value of y, we have successfully determined both unknowns in the original complex equation. This final step confirms our solution and demonstrates the power of breaking down a complex problem into simpler, manageable parts.
Therefore, the values of x and y that satisfy the equation are:
- x = 5
- y = -13
This corresponds to option D in the given choices. We have successfully navigated the complex equation and found the solution by systematically separating the real and imaginary parts and solving for the unknowns. This process highlights the importance of understanding the properties of complex numbers and applying algebraic principles to solve equations effectively. The solution, x = 5 and y = -13, is the unique set of values that makes the original equation true.
To ensure our solution is correct, let's substitute the values of x and y back into the original equation:
(5 + (-13)i) + (4 + 9i) = 9 - 4i
(5 + 4) + (-13 + 9)i = 9 - 4i
9 - 4i = 9 - 4i
The left side of the equation equals the right side, confirming that our solution is correct. This verification step is crucial in mathematics, as it provides assurance that the solution is accurate and satisfies the original problem. By substituting the values back into the equation, we have demonstrated that our solution is not only mathematically sound but also practically correct. This process reinforces the importance of checking your work and ensuring the validity of your results.
Solving complex equations may seem challenging initially, but by understanding the properties of complex numbers and applying systematic algebraic techniques, these equations can be solved with ease. In this article, we've dissected a complex equation, demonstrating the step-by-step process of finding the values of x and y that satisfy the equation. By grouping real and imaginary terms, equating corresponding parts, and solving the resulting equations, we arrived at the solution x = 5 and y = -13. This journey through complex numbers underscores the beauty and power of mathematics in unraveling intricate problems and arriving at precise solutions. The key takeaway is that complex problems can be simplified by breaking them down into manageable steps and applying fundamental principles.
