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# Number Theory/Simultaneous equations having complex numbers.

Question
Dear Prof Vijilant

Can Simultaneous equations have complex numbers and can be solved?

Example

(2+3i)x+(4-2i)y=2
(3+2i)x+(2-3i)y=1

Thanks
Prashant

Certainly.  We just follow the usual procedure.  In this case multiply (1) by 2-3i and (2) by 4-2i to get
11x + (2 - 16i)y = 4 - 6i
(16 + 2i)x + (2 - 16i)y = 4 - 2i

Subtract: (5 + 2i)x = 4i
x = 4i/(5+2i) = 4i(5 - 2i)/(5^2 + 2^2) = (8 + 20i)/29
Substitute in (1) (2 + 3i)(8 + 20i)/29 + (4 - 2i)y = 2
(-44 + 64i)/29 + (4 - 2i)y = 2
-44 + 64i +29(4 -2i)y = 58
29(4 - 2i)y = 102 -64i
y = (102 - 64i)/(29(4 - 2i)) = (102 -64i)(4 + 2i)/(29*20) =  (536 -52i)/(29*20) = (134 -13i)/145

Best wishes
vijilant

Questioner's Rating
 Rating(1-10) Knowledgeability = 10 Clarity of Response = 10 Politeness = 10 Comment Dear Prof Vijilant Thanks. Prashant

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Number Theory

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#### Vijilant

##### Expertise

Most questions on number theory, divisibility, primes, Euclidean algorithm, Fermat`s theorem, Wilson`s theorem, factorisation, euclidean algorithm, diophantine equations, Chinese remainder theorem, group theory, congruences, continued fractions.

##### Experience

Teacher of math for 53 years

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AQA Doncaster Bridge Club Danum Strings Orchestra Doncaster Conservative Club Danum Strings Orchestra Simply Voices Choir Doncaster TNS mystery shopping St Paul's Music Group Cantley

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Journal of mathematics and its applications M500 magazine

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BSc (Hons) Liverpool (Science). BA (Hons) OU (Mathematics)

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State Scholarship 1955 Highest Score in Yorkshire on OU course MST209 £50 prize First class honours in OU BA Mathematics

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I taught John Birt, former Director of the BBC in 1961. His homework book was the most perfect I have ever marked. And also the most neat. I could tell he was destined for great things. One of my classmates was the poet Roger McGough, and I have a mention in his autobiography.