Solve z(-3+4i)+z(5i-2)+3i+3=0

$$z ( - 3 + 4 i ) + z ( 5 i - 2 ) + 3 i + 3 = 0$$

Solve for z

$$z=-\frac{6}{53}+\frac{21}{53}i\approx -0.113207547+0.396226415i$$

Steps for Solving Linear Equation

Use the distributive property to multiply z by 5i-2.

$$z\left(-3+4i\right)+5iz-2z+3i+3=0$$

Combine 5iz and -2z to get \left(-2+5i\right)z.

$$z\left(-3+4i\right)+\left(-2+5i\right)z+3i+3=0$$

Combine z\left(-3+4i\right) and \left(-2+5i\right)z to get \left(-5+9i\right)z.

$$\left(-5+9i\right)z+3i+3=0$$

Subtract 3i from both sides. Anything subtracted from zero gives its negation.

$$\left(-5+9i\right)z+3=-3i$$

Subtract 3 from both sides.

$$\left(-5+9i\right)z=-3i-3$$

The equation is in standard form.

$$\left(-5+9i\right)z=-3-3i$$

Divide both sides by -5+9i.

$$\frac{\left(-5+9i\right)z}{-5+9i}=\frac{-3-3i}{-5+9i}$$

Dividing by -5+9i undoes the multiplication by -5+9i.

$$z=\frac{-3-3i}{-5+9i}$$

Divide -3-3i by -5+9i.

$$z=-\frac{6}{53}+\frac{21}{53}i$$