Solve z+3(1+v)z-8(2-i)=0

$$z + 3 ( 1 + v ) z - 8 ( 2 - i ) = 0$$

Solve for v

$$v=-\frac{4}{3}+\frac{\frac{16}{3}-\frac{8}{3}i}{z}$$

$$z\neq 0$$

Steps for Solving Linear Equation

Use the distributive property to multiply 3 by 1+v.

$$z+\left(3+3v\right)z-8\left(2-i\right)=0$$

Use the distributive property to multiply 3+3v by z.

$$z+3z+3vz-8\left(2-i\right)=0$$

Combine z and 3z to get 4z.

$$4z+3vz-8\left(2-i\right)=0$$

Multiply 8 and 2-i to get 16-8i.

$$4z+3vz-\left(16-8i\right)=0$$

Add 16-8i to both sides.

$$4z+3vz=0+\left(16-8i\right)$$

Anything plus zero gives itself.

$$4z+3vz=16-8i$$

Subtract 4z from both sides.

$$3vz=16-8i-4z$$

The equation is in standard form.

$$3zv=16-8i-4z$$

Divide both sides by 3z.

$$\frac{3zv}{3z}=\frac{16-8i-4z}{3z}$$

Dividing by 3z undoes the multiplication by 3z.

$$v=\frac{16-8i-4z}{3z}$$

Divide 16-8i-4z by 3z.

$$v=-\frac{4}{3}+\frac{\frac{16}{3}-\frac{8}{3}i}{z}$$

Solve for z

$$z=\frac{16-8i}{3v+4}$$

$$v\neq -\frac{4}{3}$$

Steps for Solving Linear Equation

Use the distributive property to multiply 3 by 1+v.

$$z+\left(3+3v\right)z-8\left(2-i\right)=0$$

Use the distributive property to multiply 3+3v by z.

$$z+3z+3vz-8\left(2-i\right)=0$$

Combine z and 3z to get 4z.

$$4z+3vz-8\left(2-i\right)=0$$

Multiply 8 and 2-i to get 16-8i.

$$4z+3vz-\left(16-8i\right)=0$$

Add 16-8i to both sides.

$$4z+3vz=0+\left(16-8i\right)$$

Anything plus zero gives itself.

$$4z+3vz=16-8i$$

Combine all terms containing z.

$$\left(4+3v\right)z=16-8i$$

The equation is in standard form.

$$\left(3v+4\right)z=16-8i$$

Divide both sides by 4+3v.

$$\frac{\left(3v+4\right)z}{3v+4}=\frac{16-8i}{3v+4}$$

Dividing by 4+3v undoes the multiplication by 4+3v.

$$z=\frac{16-8i}{3v+4}$$