thank you so much for taking the time.
Describe all solutions to z -3w - 2iw + 4iz = - 8 + 12i where z and w are complex numbers.
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We can begin by rearranging the given equation:
z - 3w - 2iw + 4iz = -8 + 12i
Simplifying the left side by grouping the real and imaginary terms of z and w separately:
z + 4iz - 3w - 2iw = -8 + 12i
Factor out z and w:
z(1+4i) - w(3+2i) = -8 + 12i
Now we can solve for one variable in terms of the other. Let's solve for z in terms of w:
z = (-8+12i+w(3+2i))/(1+4i)
To find all solutions for z and w, we need to substitute any possible values for w and solve for z.
Alternatively, we could solve for w in terms of z:
w = (-8+12i+z(1+4i))/(3+2i)
Again, we can substitute any possible values for z and solve for w.
Therefore, there are infinite solutions to this equation since we can choose any complex number for z or w, and the other variable can be calculated using the equation above.