(x+3)+i(y-2)=5+2i

3 min read Jun 16, 2024

Solving Complex Number Equations

This article will walk you through the process of solving the complex number equation (x+3) + i(y-2) = 5 + 2i.

Understanding Complex Numbers

Complex numbers are numbers that consist of a real part and an imaginary part. They are expressed in the form a + bi, where 'a' is the real part and 'b' is the imaginary part. The imaginary unit 'i' is defined as the square root of -1.

Solving the Equation

To solve the equation (x+3) + i(y-2) = 5 + 2i, we need to equate the real and imaginary parts on both sides of the equation.

Equating Real Parts:
- The real part on the left side is (x+3).
- The real part on the right side is 5.
- Therefore, we have the equation: x + 3 = 5
Equating Imaginary Parts:
- The imaginary part on the left side is (y-2).
- The imaginary part on the right side is 2.
- Therefore, we have the equation: y - 2 = 2
Solving for x and y:
- Solving the equation x + 3 = 5, we get x = 2.
- Solving the equation y - 2 = 2, we get y = 4.

Solution

Therefore, the solution to the equation (x+3) + i(y-2) = 5 + 2i is x = 2 and y = 4.

Verification

We can verify the solution by substituting the values of x and y back into the original equation:

(2 + 3) + i(4 - 2) = 5 + 2i
5 + 2i = 5 + 2i

Since both sides are equal, we have verified that the solution is correct.

Conclusion

By equating the real and imaginary parts of the complex numbers on both sides of the equation, we successfully solved for the values of x and y. This process demonstrates the fundamental principle of solving complex number equations: equating corresponding components.