document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209146>Question 1209146</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Describe all solutions to z -3w - 2iw + 4iz = -8 where z  and  w are complex numbers. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #847775 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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/cgi-bin/show-face.mpl?user=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=847775\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+-3w+-+2iw+%2B+4iz+=+-8 ALIGN=MIDDLE  ALT=\"z+-3w+-+2iw+%2B+4iz+=+-8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Let z = a+bi<BR>\n" );
document.write( "Let w = c+di<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%2Bbi%29-3%28c%2Bdi%29-2i%28c%2Bdi%29%2B4i%28a%2Bbi%29=-8 ALIGN=MIDDLE  ALT=\"%28a%2Bbi%29-3%28c%2Bdi%29-2i%28c%2Bdi%29%2B4i%28a%2Bbi%29=-8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2B4i%29%28a%2Bbi%29-%283%2B2i%29%28c%2Bdi%29=-8 ALIGN=MIDDLE  ALT=\"%281%2B4i%29%28a%2Bbi%29-%283%2B2i%29%28c%2Bdi%29=-8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a-4b-3c%2B2d%29%2B%284a%2Bb-2c-3d%29i=-8%2B0i ALIGN=MIDDLE  ALT=\"%28a-4b-3c%2B2d%29%2B%284a%2Bb-2c-3d%29i=-8%2B0i\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Equating the real and imaginary parts on both sides of the equation....<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a-4b-3c%2B2d=-8 ALIGN=MIDDLE  ALT=\"a-4b-3c%2B2d=-8\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4a%2Bb-2c-3d=0 ALIGN=MIDDLE  ALT=\"4a%2Bb-2c-3d=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "That's two equations in 4 unknowns.  The best you can do is eliminate one variable to get a single (linear) equation in three unknowns, which will have an infinite number of solutions.<br><BR>\n" );
document.write( "Solving the second equation for b and substituting in the first equation....<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=-4a%2B2c%2B3d ALIGN=MIDDLE  ALT=\"b=-4a%2B2c%2B3d\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a-4%28-4a%2B2c%2B3d%29-3c%2B2d=-8 ALIGN=MIDDLE  ALT=\"a-4%28-4a%2B2c%2B3d%29-3c%2B2d=-8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2B16a-8c-12d-3c%2B2d=-8 ALIGN=MIDDLE  ALT=\"a%2B16a-8c-12d-3c%2B2d=-8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=17a-11c-10d=-8 ALIGN=MIDDLE  ALT=\"17a-11c-10d=-8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "That last equation \"describes\" all the solutions to the given equation.<br><BR>\n" );
document.write( "For one simple solution (undoubtedly the simplest), let b=d=0, making z=a and w=c:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a-3c=-8 ALIGN=MIDDLE  ALT=\"a-3c=-8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4a-2c=0 ALIGN=MIDDLE  ALT=\"4a-2c=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Solve by elimination:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4a-2c=0 ALIGN=MIDDLE  ALT=\"4a-2c=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4a-12c=-32 ALIGN=MIDDLE  ALT=\"4a-12c=-32\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10c=32 ALIGN=MIDDLE  ALT=\"10c=32\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=3.2 ALIGN=MIDDLE  ALT=\"c=3.2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a-3%283.2%29=-8 ALIGN=MIDDLE  ALT=\"a-3%283.2%29=-8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a-9.6=-8 ALIGN=MIDDLE  ALT=\"a-9.6=-8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=1.6 ALIGN=MIDDLE  ALT=\"a=1.6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Simplest solution:<BR>\n" );
document.write( "z = 1.6+0i<BR>\n" );
document.write( "w = 3.2+0i<br><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );