Solve {l}{x+yi-2z=10}{x-y+2iz=20}{ix+3iy-(1+i)z=30}

$$\left\{ \begin{array} { l } { x + y i - 2 z = 10 } \\ { x - y + 2 i z = 20 } \\ { i x + 3 i y - ( 1 + i ) z = 30 } \end{array} \right.$$

Solve for x, y, z

$$x=3-11i$$

$$y=-3-9i$$

$$z=1-7i$$

Short Steps Using Substitution

Solve x+yi-2z=10 for x.

$$x=-iy+2z+10$$

Substitute -iy+2z+10 for x in the second and third equation.

$$-iy+2z+10-y+2iz=20 i\left(-iy+2z+10\right)+3iy-\left(1+i\right)z=30$$

Solve these equations for y and z respectively.

$$y=2z+\left(-5+5i\right) z=\left(-1+2i\right)y+\left(-20-10i\right)$$

Substitute 2z+\left(-5+5i\right) for y in the equation z=\left(-1+2i\right)y+\left(-20-10i\right).

$$z=\left(-1+2i\right)\left(2z+\left(-5+5i\right)\right)+\left(-20-10i\right)$$

Solve z=\left(-1+2i\right)\left(2z+\left(-5+5i\right)\right)+\left(-20-10i\right) for z.

$$z=1-7i$$

Substitute 1-7i for z in the equation y=2z+\left(-5+5i\right).

$$y=2\left(1-7i\right)+\left(-5+5i\right)$$

Calculate y from y=2\left(1-7i\right)+\left(-5+5i\right).

$$y=-3-9i$$

Substitute -3-9i for y and 1-7i for z in the equation x=-iy+2z+10.

$$x=-i\left(-3-9i\right)+2\left(1-7i\right)+10$$

Calculate x from x=-i\left(-3-9i\right)+2\left(1-7i\right)+10.

$$x=3-11i$$

The system is now solved.

$$x=3-11i y=-3-9i z=1-7i$$