document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1170874>Question 1170874</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help me solve this question\r<BR>\n" );
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document.write( "Given (4+2i)m - (1-i)n = -9+9i. Find the complex numbers m and n if m is the conjugate of n.\r<BR>\n" );
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document.write( "thank you. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #795756 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=795756\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Let m and n be complex numbers<BR>\n" );
document.write( "Furthermore, let n be the complex conjugate of m<BR>\n" );
document.write( "This means<BR>\n" );
document.write( "m = a+bi<BR>\n" );
document.write( "n = a-bi<BR>\n" );
document.write( "for real numbers a,b\r<BR>\n" );
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document.write( "Let's find an equivalent expression for (4+2i)m<BR>\n" );
document.write( "(4+2i)m = (4+2i)(a+bi)<BR>\n" );
document.write( "(4+2i)m = 4(a+bi)+2i(a+bi)<BR>\n" );
document.write( "(4+2i)m = 4a+4bi+2ai+2bi^2<BR>\n" );
document.write( "(4+2i)m = 4a+4bi+2ai+2b(-1)<BR>\n" );
document.write( "(4+2i)m = 4a+4bi+2ai-2b<BR>\n" );
document.write( "(4+2i) = (4a-2b)+(4bi+2ai)<BR>\n" );
document.write( "(4+2i)m = (4a-2b)+(4b+2a)i<BR>\n" );
document.write( "I'm using parenthesis to help separate out the real and imaginary parts.\r<BR>\n" );
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document.write( "Similarly, let's find (1-i)n<BR>\n" );
document.write( "(1-i)n = (1-i)(a-bi)<BR>\n" );
document.write( "(1-i)n = 1(a-bi)-i(a-bi)<BR>\n" );
document.write( "(1-i)n = a-bi-ai+bi^2<BR>\n" );
document.write( "(1-i)n = a-bi-ai+b(-1)<BR>\n" );
document.write( "(1-i)n = a-bi-ai-b<BR>\n" );
document.write( "(1-i)n = (a-b)+(-ai-bi)<BR>\n" );
document.write( "(1-i)n = (a-b)-(ai+bi)<BR>\n" );
document.write( "(1-i)n = (a-b)-(a+b)i\r<BR>\n" );
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document.write( "This means,<BR>\n" );
document.write( "(4+2i)m-(1-i)n = [(4a-2b)+(4b+2a)i] - [(a-b)-(a+b)i]<BR>\n" );
document.write( "(4+2i)m-(1-i)n = (4a-2b)+(4b+2a)i - (a-b)+(a+b)i<BR>\n" );
document.write( "(4+2i)m-(1-i)n = [(4a-2b) - (a-b)] + [(4b+2a)i+(a+b)i]<BR>\n" );
document.write( "(4+2i)m-(1-i)n = [(4a-2b) - (a-b)] + [(4b+2a)+(a+b)]i<BR>\n" );
document.write( "(4+2i)m-(1-i)n = (4a-2b - a+b) + (4b+2a+a+b)i<BR>\n" );
document.write( "(4+2i)m-(1-i)n = (3a-b) + (3a+5b)i\r<BR>\n" );
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document.write( "Equate that result with -9+9i<BR>\n" );
document.write( "(3a-b) + (3a+5b)i = -9+9i<BR>\n" );
document.write( "The corresponding parts must match up and be equal<BR>\n" );
document.write( "<ul><li>3a-b = -9</li><li>(3a+5b)i = 9i which leads to 3a+5b = 9</li></ul><BR>\n" );
document.write( "The system of equations is<BR>\n" );
document.write( "3a-b = -9<BR>\n" );
document.write( "3a+5b = 9\r<BR>\n" );
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document.write( "Let's subtract the terms straight down<BR>\n" );
document.write( "3a-3a = 0a<BR>\n" );
document.write( "-b-5b = -6b<BR>\n" );
document.write( "-9-9 = -18\r<BR>\n" );
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document.write( "We end up with -6b = -18 which solves to b = 3 after dividing both sides by -6.\r<BR>\n" );
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document.write( "We can then find 'a'<BR>\n" );
document.write( "3a-b = -9<BR>\n" );
document.write( "3a-3 = -9<BR>\n" );
document.write( "3a = -9+3<BR>\n" );
document.write( "3a = -6<BR>\n" );
document.write( "a = -6/3<BR>\n" );
document.write( "a = -2\r<BR>\n" );
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document.write( "In summary we found that<BR>\n" );
document.write( "a = -2<BR>\n" );
document.write( "b = 3\r<BR>\n" );
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document.write( "which directly leads to<BR>\n" );
document.write( "m = a+bi = -2+3i<BR>\n" );
document.write( "n = a-bi = -2-3i\r<BR>\n" );
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document.write( "Checking the answers:\r<BR>\n" );
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document.write( "(4+2i)m = (4+2i)(-2+3i)<BR>\n" );
document.write( "(4+2i)m = 4(-2+3i)+2i(-2+3i)<BR>\n" );
document.write( "(4+2i)m = -8+12i-4i+6i^2<BR>\n" );
document.write( "(4+2i)m = -8+12i-4i+6(-1)<BR>\n" );
document.write( "(4+2i)m = -8+12i-4i-6<BR>\n" );
document.write( "(4+2i)m = (-8-6)+(12-4)i<BR>\n" );
document.write( "(4+2i)m = -14+8i\r<BR>\n" );
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document.write( "(1-i)n = (1-i)(-2-3i)<BR>\n" );
document.write( "(1-i)n = 1(-2-3i)-i(-2-3i)<BR>\n" );
document.write( "(1-i)n = -2-3i+2i+3i^2<BR>\n" );
document.write( "(1-i)n = -2-3i+2i+3(-1)<BR>\n" );
document.write( "(1-i)n = -2-3i+2i-3<BR>\n" );
document.write( "(1-i)n = (-2-3)+(-3+2)i<BR>\n" );
document.write( "(1-i)n = -5-i\r<BR>\n" );
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document.write( "(4+2i)m-(1-i)n = (-14+8i) - (-5-i)<BR>\n" );
document.write( "(4+2i)m-(1-i)n = -14+8i + 5+i<BR>\n" );
document.write( "(4+2i)m-(1-i)n = (-14+5)+(8i+i)<BR>\n" );
document.write( "(4+2i)m-(1-i)n = (-14+5)+(8+1)i<BR>\n" );
document.write( "(4+2i)m-(1-i)n = -9+9i\r<BR>\n" );
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document.write( "The answers have been confirmed.\r<BR>\n" );
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document.write( "Answers:<BR>\n" );
document.write( "m = -2+3i<BR>\n" );
document.write( "n = -2-3i<BR>\n" );
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