Solve x(1+2i)+y(3+5i)=-3i

$$x ( 1 + 2 i ) + y ( 3 + 5 i ) = - 3 i$$

Solve for x

$$x=\left(-\frac{13}{5}+\frac{1}{5}i\right)y+\left(-\frac{6}{5}-\frac{3}{5}i\right)$$

Steps for Solving Linear Equation

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

$$x\left(1+2i\right)=-3i-y\left(3+5i\right)$$

Multiply -1 and 3+5i to get -3-5i.

$$x\left(1+2i\right)=-3i+\left(-3-5i\right)y$$

The equation is in standard form.

$$\left(1+2i\right)x=\left(-3-5i\right)y-3i$$

Divide both sides by 1+2i.

$$\frac{\left(1+2i\right)x}{1+2i}=\frac{\left(-3-5i\right)y-3i}{1+2i}$$

Dividing by 1+2i undoes the multiplication by 1+2i.

$$x=\frac{\left(-3-5i\right)y-3i}{1+2i}$$

Divide -3i+\left(-3-5i\right)y by 1+2i.

$$x=\left(-\frac{13}{5}+\frac{1}{5}i\right)y+\left(-\frac{6}{5}-\frac{3}{5}i\right)$$

Solve for y

$$y=\left(-\frac{13}{34}-\frac{1}{34}i\right)x+\left(-\frac{15}{34}-\frac{9}{34}i\right)$$

Steps for Solving Linear Equation

Subtract x\left(1+2i\right) from both sides.

$$y\left(3+5i\right)=-3i-x\left(1+2i\right)$$

Multiply -1 and 1+2i to get -1-2i.

$$y\left(3+5i\right)=-3i+\left(-1-2i\right)x$$

The equation is in standard form.

$$\left(3+5i\right)y=\left(-1-2i\right)x-3i$$

Divide both sides by 3+5i.

$$\frac{\left(3+5i\right)y}{3+5i}=\frac{\left(-1-2i\right)x-3i}{3+5i}$$

Dividing by 3+5i undoes the multiplication by 3+5i.

$$y=\frac{\left(-1-2i\right)x-3i}{3+5i}$$

Divide -3i+\left(-1-2i\right)x by 3+5i.

$$y=\left(-\frac{13}{34}-\frac{1}{34}i\right)x+\left(-\frac{15}{34}-\frac{9}{34}i\right)$$