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(1+3i)x+(2-5i)y=7+i

3 min read Jun 16, 2024
(1+3i)x+(2-5i)y=7+i

Solving Complex Equations: (1 + 3i)x + (2 - 5i)y = 7 + i

This article will guide you through the process of solving the complex equation (1 + 3i)x + (2 - 5i)y = 7 + i.

Understanding Complex Numbers

Before we dive into the solution, let's understand the basics of complex numbers.

A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit defined as √-1.

a is called the real part and b is called the imaginary part of the complex number.

Solving the Equation

To solve the given equation, we need to isolate the variables x and y. We can do this by separating the real and imaginary parts of the equation.

  1. Expand the equation: (1 + 3i)x + (2 - 5i)y = 7 + i x + 3ix + 2y - 5iy = 7 + i

  2. Group the real and imaginary terms: (x + 2y) + (3x - 5y)i = 7 + i

  3. Equate the real and imaginary parts: x + 2y = 7 3x - 5y = 1

  4. Solve the system of equations: We can use various methods to solve this system, such as substitution or elimination. Here, we will use elimination:

    • Multiply the first equation by 5 and the second equation by 2: 5x + 10y = 35 6x - 10y = 2

    • Add the two equations together: 11x = 37

    • Solve for x: x = 37/11

    • Substitute the value of x in either of the original equations to find y: (37/11) + 2y = 7 2y = 7 - (37/11) 2y = 34/11 y = 17/11

Solution

Therefore, the solution to the equation (1 + 3i)x + (2 - 5i)y = 7 + i is:

  • x = 37/11
  • y = 17/11

This means that the values x = 37/11 and y = 17/11 satisfy the given equation.

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