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

Jasonbradley

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Jun 17, 2024

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Solving Complex Equations: (x - 3i)(x + i) = (1 + 3i)(1 + yi)

This problem involves solving for the unknown variable, x, within a complex equation. To find the solution, we need to use the properties of complex numbers and algebraic manipulation.

1. Expand both sides of the equation:

• Left side: (x - 3i)(x + i) = x² + xi - 3ix - 3i²
• Right side: (1 + 3i)(1 + yi) = 1 + yi + 3i + 3yi²

2. Simplify using the fact that i² = -1:

• Left side: x² - 2ix + 3
• Right side: (1 + 3y) + (1 + 3)i

3. Equate the real and imaginary components:

Since two complex numbers are equal if and only if their real and imaginary parts are equal, we can equate the corresponding components:

• Real components: x² + 3 = 1 + 3y
• Imaginary components: -2x = 1 + 3

4. Solve the system of equations:

We now have two equations with two unknowns:

• Equation 1: x² + 3 = 1 + 3y
• Equation 2: -2x = 4

Solving Equation 2, we find x = -2.

Substituting x = -2 into Equation 1, we get:

• (-2)² + 3 = 1 + 3y
• 7 = 1 + 3y
• 6 = 3y
• y = 2

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