(2+i)x+(3-2i)y=-9+20i

Jasonbradley

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Jun 16, 2024

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Solving Complex Equations: (2+i)x + (3-2i)y = -9 + 20i

This article will guide you through the steps of solving the complex equation: (2+i)x + (3-2i)y = -9 + 20i. We'll break down the process into clear, understandable steps.

Understanding Complex Numbers

Before we begin, it's important to understand the basics of complex numbers:

• Complex numbers are numbers that can be expressed in the form a + bi, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit, defined as the square root of -1.
• In our equation, x and y are complex variables, meaning they can take on complex values.

Solving the Equation

1. Separate real and imaginary parts:

   We need to separate the real and imaginary parts of the equation. To do this, we distribute the complex coefficients:

   (2+i)x + (3-2i)y = -9 + 20i 2x + ix + 3y - 2iy = -9 + 20i

   Now, we group the real terms and the imaginary terms:

   (2x + 3y) + (x - 2y)i = -9 + 20i

2. Equate real and imaginary coefficients:

   For two complex numbers to be equal, their real parts and imaginary parts must be equal. This gives us two separate equations:

   • Real part equation: 2x + 3y = -9
   • Imaginary part equation: x - 2y = 20

3. Solve the system of equations:

   We now have two equations with two unknowns. We can solve this system using various methods, like substitution or elimination.

   Let's use elimination:

   • Multiply the second equation by 2: 2x - 4y = 40

   • Subtract the first equation from the modified second equation: (2x - 4y) - (2x + 3y) = 40 - (-9)

   • Simplify: -7y = 49

   • Solve for y: y = -7

   • Substitute the value of y back into either of the original equations to find x. Let's use the first equation: 2x + 3(-7) = -9

   • Simplify: 2x - 21 = -9

   • Solve for x: 2x = 12

   • Therefore, x = 6

4. Solution:

   We have found that x = 6 and y = -7.

Conclusion

The solution to the complex equation (2+i)x + (3-2i)y = -9 + 20i is x = 6 and y = -7. This process demonstrates how to solve complex equations by separating real and imaginary parts and then solving the resulting system of equations.