Question 1209146
<br>
{{{z -3w - 2iw + 4iz = -8}}}<br>
Let z = a+bi
Let w = c+di<br>
{{{(a+bi)-3(c+di)-2i(c+di)+4i(a+bi)=-8}}}<br>
{{{(1+4i)(a+bi)-(3+2i)(c+di)=-8}}}<br>
{{{(a-4b-3c+2d)+(4a+b-2c-3d)i=-8+0i}}}<br>
Equating the real and imaginary parts on both sides of the equation....<br>
{{{a-4b-3c+2d=-8}}} and {{{4a+b-2c-3d=0}}}<br>
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>
Solving the second equation for b and substituting in the first equation....<br>
{{{b=-4a+2c+3d}}}<br>
{{{a-4(-4a+2c+3d)-3c+2d=-8}}}<br>
{{{a+16a-8c-12d-3c+2d=-8}}}<br>
{{{17a-11c-10d=-8}}}<br>
That last equation "describes" all the solutions to the given equation.<br>
For one simple solution (undoubtedly the simplest), let b=d=0, making z=a and w=c:<br>
{{{a-3c=-8}}}
{{{4a-2c=0}}}<br>
Solve by elimination:<br>
{{{4a-2c=0}}}
{{{4a-12c=-32}}}
{{{10c=32}}}
{{{c=3.2}}}
{{{a-3(3.2)=-8}}}
{{{a-9.6=-8}}}
{{{a=1.6}}}<br>
Simplest solution:
z = 1.6+0i
w = 3.2+0i<br>