Question

Solve for all possible values of the real numbers \(x\) and \(y\) in the following equations. $$\frac{x+i y+2+3 i}{2 x+2 i y-3}=i+2$$

Short answer

x = \(\frac{8}{3}\), y = \(\frac{4}{3}\)

Step-by-step solution

Simplify the Right-Hand Side

Express the right-hand side of the equation as a complex number: i + 2 = 2 + i.

Equate Numerator and Denominator

Write the original equation:\[ \frac{x + i y + 2 + 3i}{2x + 2i y - 3} = 2 + i \]For this fraction to equal the complex number \(2 + i\), the numerator and denominator must be proportional to \(2 + i\).

Separate Real and Imaginary Parts

Equate the numerator to the right-hand side after multiplying by the denominator on both sides:\[ x + i y + 2 + 3i = (2 + i)(2x + 2iy - 3) \]Now expand the right-hand side:\[ x + i y + 2 + 3i = 4x + 4iy - 6 + 2i - 3i \]Simplify:\[ x + i y + 2 + 3i = 4x + i(4y - 1) - 6 \]

Compare Real and Imaginary Parts

Compare the real parts on both sides of the equation:\[ x + 2 = 4x - 6 \]and the imaginary parts:\[ y + 3 = 4y - 1 \]

Solve for x

Solving the real part equation:\[ x + 2 = 4x - 6 \]\[ x - 4x = -6 - 2 \]\[ -3x = -8 \]\[ x = \frac{8}{3} \]

Solve for y

Solving the imaginary part equation:\[ y + 3 = 4y - 1 \]\[ 3 + 1 = 4y - y \]\[ 4 = 3y \]\[ y = \frac{4}{3} \]

Key concepts

Understanding Complex Numbers

Complex numbers are numbers that have both a real part and an imaginary part. The real part is a regular number, while the imaginary part is a multiple of the imaginary unit, denoted as \(i\). The imaginary unit \(i\) is defined by the property \(i^2 = -1\). A complex number can be written in the form \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. Understanding how to manipulate and solve equations involving complex numbers is crucial in various fields of science and engineering.
The given equation in the exercise involves a fraction that contains complex numbers. You can think of the fraction as a division of complex numbers, and understanding how these parts interact is key to solving the equation.

Solving Equations Involving Complex Numbers

Solving equations that include complex numbers requires a systematic approach where you separate the complex equation into its real and imaginary parts. In the given exercise, our goal is to find the real numbers \(x\) and \(y\) that satisfy the equation. The step-by-step solution showed how we can simplify and manipulate the given equation to make it easier to solve. Here's a quick recap:

• Simplify the right-hand side of the equation to express it as a complex number.
• Rewrite the original equation and aim to make the numerator and denominator proportional.
• Separate the real and imaginary parts by comparing corresponding parts on both sides of the equation.
• Solve the resulting equations for \(x\) and \(y\).

By following these steps, we determine the values of \(x\) and \(y\) that make the original complex equation true.

Proportionality in Complex Numbers

Proportionality in complex numbers is a concept where one complex number can be expressed as a multiple of another. In the exercise, we sought to equate the fraction involving complex numbers to another complex number. This meant the numerator had to be a proportionate form of the denominator.
When we multiplied one side of the equation by the denominator on the other side, we effectively created two scenarios for comparison: the real parts and the imaginary parts. By treating them as separate proportions, we matched corresponding parts to form solvable equations in real numbers. This technique is essential when dealing with equations that contain complex numbers.

Separating Real and Imaginary Parts

Separating the real and imaginary parts of a complex equation is crucial for finding solutions. In this exercise, we had a complex equation that we broke down by equating the real parts and the imaginary parts on either side.
For instance, consider the expression \(a + bi = c + di\):

• The real part is \(a\) and \(c\), which gives the equation: \(a = c\).
• The imaginary part is \(bi\) and \(di\), giving: \(b = d\).

By comparing these parts, we can solve for the individual variables in our original equation. In the given exercise, by comparing the real parts, we formed the equation \(x + 2 = 4x - 6\). By comparing the imaginary parts, we had \(y + 3 = 4y - 1\). These comparisons allowed us to isolate and solve for \(x\) and \(y\) separately.