(3-2i)z+(4+5i)=7+3i

Jasonbradley

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Jun 16, 2024

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Solving Complex Equation: (3-2i)z + (4+5i) = 7+3i

This article will guide you through the steps of solving the complex equation (3-2i)z + (4+5i) = 7+3i for the unknown complex number z.

Understanding Complex Numbers

A complex number is a number of the form a + bi, where a and b are real numbers, and i is the imaginary unit, defined as the square root of -1.

Solving the Equation

1. Isolate the term with 'z': Subtract (4+5i) from both sides of the equation: (3-2i)z = (7+3i) - (4+5i) Simplifying the right side: (3-2i)z = 3 - 2i

2. Divide both sides by (3-2i): To isolate z, divide both sides by (3-2i). Remember that dividing by a complex number involves multiplying both the numerator and denominator by the conjugate of the divisor.

   The conjugate of (3-2i) is (3+2i).

   Therefore: z = (3 - 2i) / (3 - 2i) * (3 + 2i) / (3 + 2i)

3. Simplify: Expanding the numerator and denominator: z = (9 + 6i - 6i - 4i²) / (9 + 6i - 6i - 4i²) Recall that i² = -1 z = (9 + 4) / (9 + 4) z = 13/13 z = 1

Solution

Therefore, the solution to the complex equation (3-2i)z + (4+5i) = 7+3i is z = 1.