Question

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

Short answer

The solution is any real number where \(x = y\).

Step-by-step solution

Identify the given equation

The given equation is \(x + i y = y + i x\). We need to solve for the real numbers \(x\) and \(y\).

Separate real and imaginary parts

To solve for \(x\) and \(y\), separate the real and imaginary components of the equation. The real part is \(x\) on the left-hand side and \(y\) on the right-hand side. The imaginary part is \(i y\) on the left-hand side and \(i x\) on the right-hand side. So the equation can be written as: Real part: \(x = y\) Imaginary part: \(y = x\)

Analyze the separated parts

Since both the real part and the imaginary part give us the same equation \(x = y\), we can deduce that \(x\) must equal \(y\) for both parts to be satisfied. Therefore, the solution to the given equation is any real number where \(x = y\).

Key concepts

Real Part

In complex numbers, every number has a real part and an imaginary part.
The real part of a complex number is the component that does not involve the imaginary unit \(i\).
For example, in the complex number \(a + bi\), \a\ is the real part.
This is because it is the term without the \(i\).
To solve equations involving complex numbers, it's crucial to separate the real parts from the imaginary parts.
In our given equation, \(x + i y = y + i x\), the real parts are \x\ and \y\.
Since both sides of the equation must be equal, we equate the real parts: \(x = y\).

Imaginary Part

The imaginary part of a complex number is the component that involves the imaginary unit \(i\).
The imaginary unit \(i\) is defined by the property \(i^2 = -1\).
For instance, in the complex number \(a + bi\), \bi\ is the imaginary part, where \(b\) is a real number coefficient.
In our example, the imaginary parts of the equation \(x + i y = y + i x\) are \(i y\ and \(i x\).
Just like with the real parts, the imaginary parts must be equal for the equation to hold true.
Thus, we set \i y = i x\).
After simplifying by dividing both sides by \(i\), we also get \(x = y\).

Solving Equations

To solve equations with complex numbers, break down the equation into its real and imaginary parts.
This allows us to handle each component separately.
Start by comparing the real parts and setting them equal to each other.
Do the same for the imaginary parts.
In the equation \(x + i y = y + i x\), we first identified the real and imaginary parts.
Both comparisons led us to \(x = y\).
This means that any real numbers for \(x\) and \(y\) which satisfy \(x = y\) are solutions to the equation.
Practicing this method will make it easier to solve more complex equations in the future.