Solving Complex Equations: (52i)7=x(3+yi)
This article will guide you through solving the complex equation (52i)7=x(3+yi). We will use the properties of complex numbers and algebraic manipulations to find the values of x and y.
Understanding 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. The real part of a complex number is 'a' and the imaginary part is 'b'.
Solving the Equation

Simplify both sides of the equation:
 (52i)7 = 2  2i
 x(3+yi) = (x3)  yi

Equate the real and imaginary parts: For two complex numbers to be equal, their real and imaginary parts must be equal. Therefore:
 2 = x  3
 2 = y

Solve for x and y:
 x = 2 + 3 = 1
 y = 2
Solution
Therefore, the solution to the equation (52i)7=x(3+yi) is:
 x = 1
 y = 2
This means that the equation is true when we substitute x = 1 and y = 2.
Conclusion
This example demonstrates how to solve a complex equation by separating the real and imaginary components. By applying the rules of complex numbers and basic algebra, we can find the values of the unknown variables.