(2+3i)z+4-3i=13+4i

3 min read Jun 16, 2024

Solving Complex Equations: (2+3i)z + 4 - 3i = 13 + 4i

This article will guide you through the steps involved in solving the complex equation: (2 + 3i)z + 4 - 3i = 13 + 4i.

Before we start, let's recap what complex numbers are:

Complex numbers are numbers that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, defined as the square root of -1.
Real numbers are a subset of complex numbers where the imaginary part is zero (b=0).
Imaginary numbers are complex numbers where the real part is zero (a=0).

Let's solve the equation step-by-step:

Isolate the term with 'z': Subtract 4 - 3i from both sides of the equation: (2 + 3i)z = 13 + 4i - (4 - 3i) (2 + 3i)z = 9 + 7i
Divide both sides by (2 + 3i): To isolate 'z', divide both sides by (2 + 3i). Remember, division by complex numbers involves multiplying both the numerator and denominator by the conjugate of the denominator. The conjugate of (2 + 3i) is (2 - 3i). z = (9 + 7i) * (2 - 3i) / (2 + 3i) * (2 - 3i)
Simplify: Expand the multiplication: z = (18 - 27i + 14i - 21i^2) / (4 - 9i^2)
Substitute i^2 = -1: z = (18 - 13i + 21) / (4 + 9) z = (39 - 13i) / 13
Express in the form a + bi: z = (39/13) - (13/13)i z = 3 - i

Therefore, the solution to the complex equation (2 + 3i)z + 4 - 3i = 13 + 4i is z = 3 - i.