Solve z=3i(z-i)-5z

$$z = 3 i ( z - i ) - 5 z$$

Solve for z

$$z=\frac{2}{5}+\frac{1}{5}i=0.4+0.2i$$

Steps for Solving Linear Equation

Use the distributive property to multiply 3i by z-i.

$$z=3iz+3-5z$$

Combine 3iz and -5z to get \left(-5+3i\right)z.

$$z=\left(-5+3i\right)z+3$$

Subtract \left(-5+3i\right)z from both sides.

$$z-\left(-5+3i\right)z=3$$

Combine z and \left(5-3i\right)z to get \left(6-3i\right)z.

$$\left(6-3i\right)z=3$$

Divide both sides by 6-3i.

$$z=\frac{3}{6-3i}$$

Multiply both numerator and denominator of \frac{3}{6-3i} by the complex conjugate of the denominator, 6+3i.

$$z=\frac{3\left(6+3i\right)}{\left(6-3i\right)\left(6+3i\right)}$$

Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$z=\frac{3\left(6+3i\right)}{6^{2}-3^{2}i^{2}}$$

By definition, i^{2} is -1. Calculate the denominator.

$$z=\frac{3\left(6+3i\right)}{45}$$

Multiply 3 times 6+3i.

$$z=\frac{3\times 6+3\times \left(3i\right)}{45}$$

Do the multiplications in 3\times 6+3\times \left(3i\right).

$$z=\frac{18+9i}{45}$$

Divide 18+9i by 45 to get \frac{2}{5}+\frac{1}{5}i.

$$z=\frac{2}{5}+\frac{1}{5}i$$