(1+i)(x+iy)=2-5i

Jasonbradley

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Jun 16, 2024

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Solving the Complex Equation: (1+i)(x+iy) = 2-5i

This article will walk you through the process of solving the complex equation (1+i)(x+iy) = 2-5i. We will use the fundamental properties of complex numbers to isolate the real and imaginary components and find the values of x and y.

Understanding Complex Numbers

Before we dive into the solution, let's recall some key points about complex numbers:

• Form: Complex numbers are expressed in the form a + bi, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit (√-1).
• Multiplication: When multiplying complex numbers, we distribute just like we would with any binomial: (a + bi)(c + di) = ac + adi + bci + bdi² = (ac - bd) + (ad + bc)i.

Solving the Equation

1. Expand the left side: (1 + i)(x + iy) = 1(x) + 1(iy) + i(x) + i(iy) = (x - y) + (x + y)i

2. Equate real and imaginary components: Since the equation states that the left side is equal to 2-5i, we can equate the real and imaginary components:

   • x - y = 2
   • x + y = -5

3. Solve the system of equations: We now have two linear equations with two unknowns. We can solve this system using various methods, such as substitution or elimination. Let's use elimination:

   • Add the two equations together: (x - y) + (x + y) = 2 - 5 2x = -3 x = -3/2

   • Substitute the value of x back into either of the original equations to solve for y. Let's use the first equation: (-3/2) - y = 2 -y = 7/2 y = -7/2

Solution

Therefore, the solution to the complex equation (1+i)(x+iy) = 2-5i is:

• x = -3/2
• y = -7/2

This can be expressed in complex number form as: x + iy = -3/2 - 7/2i