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Solving the Equation (3+i)x(1-2i)y + 7i = 0
This article will guide you through the process of solving the complex equation (3+i)x(1-2i)y + 7i = 0. We will break down the steps and explain the concepts involved.
Understanding Complex Numbers
Before we begin, let's quickly recap some key aspects of complex numbers:
- Complex numbers are numbers of the form a + bi, where 'a' and 'b' are real numbers and 'i' is the imaginary unit (i² = -1).
- Multiplication of complex numbers follows the distributive property, similar to multiplying binomials.
Solving the Equation
Our equation is (3+i)x(1-2i)y + 7i = 0. To solve for x and y, we will follow these steps:
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Expand the multiplication:
- First, multiply the terms (3+i) and (1-2i).
- We get (3 + i)(1 - 2i) = 3 - 6i + i - 2i² = 3 - 5i + 2 = 5 - 5i.
- Our equation now becomes (5-5i)xy + 7i = 0.
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Isolate the xy term:
- Subtract 7i from both sides: (5-5i)xy = -7i.
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Divide both sides by (5-5i):
- This step involves dividing by a complex number. To do this, we multiply both the numerator and denominator by the conjugate of the denominator.
- The conjugate of (5-5i) is (5+5i).
- We get: xy = (-7i)(5+5i) / (5-5i)(5+5i)
- Simplifying the equation: xy = (-35i - 35i²) / (25 + 25) = (-35i + 35) / 50 = (7 - 7i) / 10.
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Express the solution:
- Our final result shows that the product xy is a complex number: xy = (7/10) - (7/10)i.
- This means that x and y can be any pair of complex numbers whose product equals (7/10) - (7/10)i.
Conclusion
We have successfully solved the equation (3+i)x(1-2i)y + 7i = 0. The solution shows that there are infinitely many possible values for x and y, as long as their product equals (7/10) - (7/10)i. Remember that complex numbers offer a rich and fascinating area of mathematics, with applications in various fields.