(3-2i)(x+yi)=2(x-2yi)+2i-1

Jasonbradley

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Jun 16, 2024

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Solving the Complex Equation: (3-2i)(x+yi)=2(x-2yi)+2i-1

This article explores the solution to the complex equation (3-2i)(x+yi)=2(x-2yi)+2i-1. We will utilize the properties of complex numbers and algebraic manipulation to find the values of x and y that satisfy this equation.

Expanding and Simplifying

First, let's expand both sides of the equation:

• Left-hand side: (3-2i)(x+yi) = 3x + 3yi - 2ix - 2i²y = (3x + 2y) + (3y - 2x)i
• Right-hand side: 2(x-2yi)+2i-1 = 2x - 4yi + 2i - 1 = (2x - 1) + (-4y + 2)i

Now, we can equate the real and imaginary components of both sides:

• Real component: 3x + 2y = 2x - 1
• Imaginary component: 3y - 2x = -4y + 2

Solving the System of Equations

We now have a system of two equations with two unknowns. Solving for x and y:

1. From the real component: x = -1 - 2y

2. Substituting this value of x into the imaginary component: 3y - 2(-1 - 2y) = -4y + 2 3y + 2 + 4y = -4y + 2 11y = 0 y = 0

3. Substituting the value of y back into the equation for x: x = -1 - 2(0) x = -1

Therefore, the solution to the equation (3-2i)(x+yi)=2(x-2yi)+2i-1 is x = -1 and y = 0.

Verification

To verify our solution, let's substitute these values back into the original equation:

(3-2i)(-1+0i) = 2(-1-0i)+2i-1 -3 + 2i = -2 + 2i - 1

This confirms that both sides of the equation are equal, validating our solution.

In conclusion, we have successfully found the solution for the complex equation (3-2i)(x+yi)=2(x-2yi)+2i-1. By expanding, separating real and imaginary components, and solving the resulting system of equations, we determined that x = -1 and y = 0 are the values that satisfy the equation.