Question

Solve for all possible values of the real numbers \(x\) and \(y\) in the following equations. $$x+i y=3 i-4$$

Short answer

The values are \(x = -4\) and \(y = 3\).

Step-by-step solution

Understand the Equation

Given the equation: \(x + i y = 3i - 4\) Recall that both sides of the equation must be equal in real and imaginary parts separately.

Separate the Real and Imaginary Parts

Since the given equation includes both real and imaginary parts, we need to equate the real parts and the imaginary parts separately. Real part: \(x = -4\) Imaginary part: \(y = 3\)

Solve for x and y

From the separation in Step 2, we have: \(x = -4\) and \(y = 3\)

Verify the Solution

Substitute the values found back into the original equation to verify: \(x + i y =-4 + 3i = 3i - 4\)Both sides of the equation match, confirming the solution is correct.

Key concepts

Real and Imaginary Parts

When dealing with complex numbers, it's important to understand that they have two distinct components: a real part and an imaginary part. A complex number is usually written in the form of \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. The imaginary unit \(i\) is defined such that \(i^2 = -1\).

In our specific exercise, the complex number \((x + iy)\) has to be split into its real part \((x)\) and its imaginary part \((iy)\). When the equation involves complex numbers on both sides, both the real and imaginary parts on each side must be equal. This allows for the separation of the equation into two simpler ones.

Knowing how to separate these parts is essential for solving equations that involve complex numbers. This concept is fundamental in many areas of math and engineering.

Solving Equations

Solving equations with complex numbers involves separating and equating the real and imaginary parts as two separate equations. In the given exercise, the equation \(x + iy = 3i - 4\), we can identify the real part as \(-4\) and the imaginary part as \(3i\).

Here's how we break down the steps:

• Step 1: Write down the equation given: \(x + iy = 3i - 4\).
• Step 2: Separate the real part and the imaginary part. This gives us: \(x = -4\) and \(iy = 3i\).
• Step 3: Solve each part individually. For the real part: \(x = -4\), and for the imaginary part: \(y = 3\).

By focusing on the real and imaginary parts separately, you simplify the solving process immensely. Once separated, these equations often become straightforward to solve individually.

Verification of Solutions

After solving for the unknowns in the equation, it's crucial to verify that your solutions satisfy the original equation. Verification ensures no mistakes were made during the solving process.

In our exercise, we found \(x = -4\) and \(y = 3\). To verify:

• Substitute \(x\) and \(y\) back into the original equation: \(x + iy = -4 + 3i\).
• Check if it matches the right side of the original equation: \(3i - 4\).

As both sides match, this confirms our solution is correct. Verification serves as a final check to authenticate your results. This step is critical in mathematics to ensure the solution holds true under the original constraints.