Solve the following system of linear equations with complex coefficients for omega and z: 2iz + (3−2i)omega = 1+i and (1−2i)z + (3+2i)omega = 5+6i.

$$2iz + (3-2i)\omega = 1+i \quad \text{(1)}$$ $$(1-2i)z + (3+2i)\omega = 5+6i \quad \text{(2)}$$

Multiply Equation (1) by $(1-2i)$ and Equation (2) by $2i$.
Multiply Equation (1) by $(1-2i)$:

$$(1-2i)(2iz) + (1-2i)(3-2i)\omega = (1-2i)(1+i)$$ $$(2i - 4i^2)z + (3 - 2i - 6i + 4i^2)\omega = (1 + i - 2i - 2i^2)$$ $$(4+2i)z + (-1-8i)\omega = (3-i) \quad \text{(3)}$$

Multiply Equation (2) by $2i$:

$$(2i)(1-2i)z + (2i)(3+2i)\omega = (2i)(5+6i)$$ $$(2i - 4i^2)z + (6i + 4i^2)\omega = (10i + 12i^2)$$ $$(4+2i)z + (-4+6i)\omega = (-12+10i) \quad \text{(4)}$$

$$[(4+2i)z + (-1-8i)\omega] - [(4+2i)z + (-4+6i)\omega] = (3-i) - (-12+10i)$$ $$(-1-8i+4-6i)\omega = 3-i+12-10i$$ $$(3-14i)\omega = 15-11i$$

$$\omega = \frac{15-11i}{3-14i}$$

Multiply the numerator and denominator by the conjugate of the denominator, $3+14i$:

$$\omega = \frac{(15-11i)(3+14i)}{(3-14i)(3+14i)}$$ $$\omega = \frac{15(3) + 15(14i) - 11i(3) - 11i(14i)}{3^2 - (14i)^2}$$ $$\omega = \frac{45 + 210i - 33i - 154i^2}{9 - 196i^2}$$ $$\omega = \frac{45 + 177i + 154}{9 + 196}$$ $$\omega = \frac{199 + 177i}{205}$$ $$\boxed{\omega = \frac{199}{205} + \frac{177}{205}i}$$

$$2iz + (3-2i)\omega = 1+i$$ $$2iz = 1+i - (3-2i)\omega$$ $$2iz = 1+i - (3-2i)\left(\frac{199+177i}{205}\right)$$ $$2iz = \frac{205(1+i) - (3-2i)(199+177i)}{205}$$

First, calculate $(3-2i)(199+177i)$:

$$(3-2i)(199+177i) = 3(199) + 3(177i) - 2i(199) - 2i(177i)$$ $$= 597 + 531i - 398i - 354i^2$$ $$= 597 + 133i + 354$$ $$= 951 + 133i$$

Now substitute this back:

$$2iz = \frac{205+205i - (951+133i)}{205}$$ $$2iz = \frac{205+205i - 951 - 133i}{205}$$ $$2iz = \frac{-746 + 72i}{205}$$

$$z = \frac{-746 + 72i}{205 \times 2i}$$ $$z = \frac{-746 + 72i}{410i}$$

Multiply the numerator and denominator by $-i$:

$$z = \frac{(-746 + 72i)(-i)}{410i(-i)}$$ $$z = \frac{746i - 72i^2}{-410i^2}$$ $$z = \frac{746i + 72}{410}$$ $$z = \frac{72}{410} + \frac{746}{410}i$$

Simplify the fractions:

$$z = \frac{36}{205} + \frac{373}{205}i$$ $$\boxed{z = \frac{36}{205} + \frac{373}{205}i}$$