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(x+iy)(2-3i)=4+i

2 min read Jun 17, 2024
(x+iy)(2-3i)=4+i

Solving the Complex Equation: (x + iy)(2 - 3i) = 4 + i

This article explores the process of solving the complex equation (x + iy)(2 - 3i) = 4 + i, where x and y are real numbers and i is the imaginary unit (i² = -1).

Expanding and Equating Real and Imaginary Parts

  1. Expand the left-hand side: (x + iy)(2 - 3i) = 2x + 2iy - 3ix - 3iy² = (2x + 3y) + (-3x + 2y)i

  2. Equate the real and imaginary parts: We have (2x + 3y) + (-3x + 2y)i = 4 + i. This gives us two equations:

    • 2x + 3y = 4
    • -3x + 2y = 1

Solving the System of Equations

We now have a system of two linear equations with two unknowns. There are various ways to solve this system, such as substitution or elimination. Let's use elimination:

  1. Multiply the first equation by 3 and the second equation by 2:

    • 6x + 9y = 12
    • -6x + 4y = 2

  2. Add the two equations:

    • 13y = 14

  3. Solve for y:

    • y = 14/13

  4. Substitute the value of y back into either of the original equations to solve for x:

    • 2x + 3(14/13) = 4
    • 2x = 4 - 42/13
    • 2x = 10/13
    • x = 5/13

Solution

Therefore, the solution to the equation (x + iy)(2 - 3i) = 4 + i is x = 5/13 and y = 14/13.

This means the complex number (x + iy) that satisfies the equation is (5/13 + 14/13 i).

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