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Solving Complex Equations: (5-2i)-7=x-(3+yi)
This article will guide you through solving the complex equation (5-2i)-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
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Simplify both sides of the equation:
- (5-2i)-7 = -2 - 2i
- x-(3+yi) = (x-3) - yi
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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
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Solve for x and y:
- x = -2 + 3 = 1
- y = 2
Solution
Therefore, the solution to the equation (5-2i)-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.