document.write( "Question 1209146: Describe all solutions to z -3w - 2iw + 4iz = -8 where z and w are complex numbers. \n" ); document.write( "

Algebra.Com's Answer #847777 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "Describe all solutions to z -3w - 2iw + 4iz = -8 where z and w are complex numbers.
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document.write( "Your starting equation is\r\n" );
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document.write( "    z -3w - 2iw + 4iz = -8, \r\n" );
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document.write( "where z  and  w are complex numbers/variables.\r\n" );
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document.write( "In the equation, collect and combine like terms.  You will get\r\n" );
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document.write( "    (z+4iz) - (3w+2iw) = -8,\r\n" );
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document.write( "    (1+4i)z - (3+2i)w = -8,\r\n" );
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document.write( "    (1+4i)z = -8 + (3+2i)w,\r\n" );
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document.write( "    z =  + .\r\n" );
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document.write( "Thus, in this case, there are infinitely many possible solutions.\r\n" );
document.write( "\"w\"  can be any complex number, and then  \"z\"  is expressed via  \"w\"  by this formula.\r\n" );
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document.write( "It is the full description of the solution set.\r\n" );
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document.write( "It is similar to the regular real case, when you are given one linear equation \r\n" );
document.write( "for two unknown, and you are asked to describe all possible solutions.\r\n" );
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document.write( "Then (in the regular case) one unknown is a free variable (as \"w\" in this case), \r\n" );
document.write( "while the second unknown is a linear function of the first variable.\r\n" );
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\n" ); document.write( "\n" ); document.write( "It is what you need to understand and what they want to get / (to hear) from you as your answer.\r
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