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(4-3i)x(2+5i)y=6-11i

3 min read Jun 16, 2024
(4-3i)x(2+5i)y=6-11i

Solving Complex Equation: (4-3i)x(2+5i)y = 6-11i

This article explores how to solve the complex equation (4-3i)x(2+5i)y = 6-11i for the unknown variables x and y.

Understanding Complex Numbers

Complex numbers are numbers of the form a + bi, where a and b are real numbers and i is the imaginary unit, defined as the square root of -1.

Solving the Equation

  1. Expand the left side:

    The first step is to expand the left side of the equation by multiplying the complex numbers:

    (4-3i)x(2+5i)y = (8 + 20i - 6i - 15i²)xy = (8 + 14i + 15)xy = (23 + 14i)xy

  2. Equate Real and Imaginary Parts:

    Since the equation is an equality between complex numbers, the real and imaginary parts on both sides must be equal. We can write this as:

    • Real part: 23xy = 6
    • Imaginary part: 14xy = -11

  3. Solve for x and y:

    We now have a system of two equations with two unknowns. We can solve for x and y using various methods like substitution or elimination.

    • Using Elimination:

      • Multiply the first equation by -14 and the second equation by 23: -322xy = -84 322xy = -253
      • Adding these equations gives: 0 = -337, which is a contradiction.

    • Using Substitution:

      • Solve the first equation for xy: xy = 6/23
      • Substitute this value of xy into the second equation: 14(6/23) = -11
      • This leads to 84/23 = -11, which is again a contradiction.

Conclusion

The contradiction obtained from both methods indicates that there is no solution to the equation (4-3i)x(2+5i)y = 6-11i for the real variables x and y. This means no pair of real numbers x and y can satisfy the given equation.

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