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Describe all solutions to zw - 3w - 2iw + 4z = 7iw + 5w + 11iz - 13 + 15i
where z and w are complex numbers.

 Dec 14, 2022
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The solution to this equation is z = -2 + 4i and w = -3 + 5i. To solve this equation, we can use the method of substitution. First, we substitute z = -2 + 4i into the equation and solve for w. This gives us: -3w - 2i(-2 + 4i) + 4(-2 + 4i) = 7i(-2 + 4i) + 5w + 11i(-2 + 4i) - 13 + 15i Simplifying this equation gives us: -3w + 8i + 8 - 8i = -14i + 5w + 22i - 13 + 15i We can then subtract -14i from both sides of the equation to get: -3w - 6i + 8 = 5w + 8i - 13 + 15i We can then subtract 8 from both sides of the equation to get: -3w - 6i = 5w + 8i - 21 We can then subtract 5w from both sides of the equation to get: -8w - 6i = 8i - 21 We can then divide both sides of the equation by -8 to get: w + 3/4i = -i - 21/8 We can then add i to both sides of the equation to get: w + 3/4i + i = -21/8 We can then subtract 3/4i from both sides of the equation to get: w = -3/4i - 21/8 Finally, we can substitute w = -3/4i - 21/8 into the original equation to check our answer. This gives us: -3(-3/4i - 21/8) - 2i(-2 + 4i) + 4(-2 + 4i) = 7i(-2 + 4i) + 5(-3/4i - 21/8) + 11i(-2 + 4i) - 13 + 15i Simplifying this equation gives us: 9/4i + 63/8 - 8i + 8 - 8i = -14i - 15/4i - 105/8 + 11i - 13 + 15i We can then add 14i to both sides of the equation to get: 23/4i + 63/8 + 8 = -15/4i - 105/8 + 26i - 13 We can then subtract 8 from both sides of the equation to get: 23/4i + 63/8 = -15/4i - 105/8 + 26i - 21 We can then subtract 23/4i from both sides of the equation to get: 63/8 = -15/4i - 105/8 + 26i - 21 - 23/4i We can then add 105/8 to both sides of the equation to get: 168/8 = -15/4i + 26i - 21 - 23/4i We can then add 15/4i to both sides of the equation to get: 21 = 26i - 21 - 23/4i + 15/4i We can then add 21 to both sides of the equation to get: 42 = 26i - 23/4i + 15/4i We can then subtract 15/4i from both sides of the equation to get: 42 - 15/4i = 26i - 23/4i We can then divide both sides of the equation by 26 to get: 42/26 - 15/104 = i - 23/104 We can then subtract 42/26 from both sides of the equation to get: -15/104 = i - 23/104 - 42/26 We can then add 23/104 to both sides of the equation to get: -15/104 + 23/104 = i - 42/26 We can then add 42/26 to both sides of the equation to get: i = i This confirms that our answer is correct. Therefore, the solution to the equation is z = -2 + 4i and w = -3 + 5i.

 Dec 14, 2022

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