Solve (3x-i)(2+i)+(x-iy)(1+2i)=5+6i

$$( 3 x - i ) ( 2 + i ) + ( x - i y ) ( 1 + 2 i ) = 5 + 6 i$$

Solve for x

$$x=\left(-\frac{9}{74}+\frac{17}{74}i\right)y+\left(\frac{34}{37}+\frac{18}{37}i\right)$$

Steps for Solving Linear Equation

Use the distributive property to multiply 3x-i by 2+i.

$$\left(6+3i\right)x+\left(1-2i\right)+\left(x-iy\right)\left(1+2i\right)=5+6i$$

Subtract 1-2i from both sides.

$$\left(6+3i\right)x+\left(x-iy\right)\left(1+2i\right)=5+6i-\left(1-2i\right)$$

Subtract 1-2i from 5+6i to get 4+8i.

$$\left(6+3i\right)x+\left(x-iy\right)\left(1+2i\right)=4+8i$$

Use the distributive property to multiply x-iy by 1+2i.

$$\left(6+3i\right)x+\left(1+2i\right)x+\left(2-i\right)y=4+8i$$

Combine \left(6+3i\right)x and \left(1+2i\right)x to get \left(7+5i\right)x.

$$\left(7+5i\right)x+\left(2-i\right)y=4+8i$$

Subtract \left(2-i\right)y from both sides.

$$\left(7+5i\right)x=4+8i-\left(2-i\right)y$$

The equation is in standard form.

$$\left(7+5i\right)x=\left(-2+i\right)y+\left(4+8i\right)$$

Divide both sides by 7+5i.

$$\frac{\left(7+5i\right)x}{7+5i}=\frac{\left(-2+i\right)y+\left(4+8i\right)}{7+5i}$$

Dividing by 7+5i undoes the multiplication by 7+5i.

$$x=\frac{\left(-2+i\right)y+\left(4+8i\right)}{7+5i}$$

Divide 4+8i+\left(-2+i\right)y by 7+5i.

$$x=\left(-\frac{9}{74}+\frac{17}{74}i\right)y+\left(\frac{34}{37}+\frac{18}{37}i\right)$$

Solve for y

$$y=\left(-\frac{9}{5}-\frac{17}{5}i\right)x+4i$$

Steps for Solving Linear Equation

Use the distributive property to multiply 3x-i by 2+i.

$$\left(6+3i\right)x+\left(1-2i\right)+\left(x-iy\right)\left(1+2i\right)=5+6i$$

Subtract \left(6+3i\right)x from both sides.

$$1-2i+\left(x-iy\right)\left(1+2i\right)=5+6i-\left(6+3i\right)x$$

Subtract 1-2i from both sides.

$$\left(x-iy\right)\left(1+2i\right)=5+6i-\left(6+3i\right)x-\left(1-2i\right)$$

Use the distributive property to multiply x-iy by 1+2i.

$$\left(1+2i\right)x+\left(2-i\right)y=5+6i-\left(6+3i\right)x-\left(1-2i\right)$$

Multiply -1 and 6+3i to get -6-3i.

$$\left(1+2i\right)x+\left(2-i\right)y=5+6i+\left(-6-3i\right)x-\left(1-2i\right)$$

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

$$\left(1+2i\right)x+\left(2-i\right)y=5+6i+\left(-6-3i\right)x+\left(-1+2i\right)$$

Do the additions in 5+6i+\left(-1+2i\right).

$$\left(1+2i\right)x+\left(2-i\right)y=\left(-6-3i\right)x+4+8i$$

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

$$\left(2-i\right)y=\left(-6-3i\right)x+4+8i-\left(1+2i\right)x$$

Combine \left(-6-3i\right)x and \left(-1-2i\right)x to get \left(-7-5i\right)x.

$$\left(2-i\right)y=\left(-7-5i\right)x+4+8i$$

The equation is in standard form.

$$\left(2-i\right)y=\left(-7-5i\right)x+\left(4+8i\right)$$

Divide both sides by 2-i.

$$\frac{\left(2-i\right)y}{2-i}=\frac{\left(-7-5i\right)x+\left(4+8i\right)}{2-i}$$

Dividing by 2-i undoes the multiplication by 2-i.

$$y=\frac{\left(-7-5i\right)x+\left(4+8i\right)}{2-i}$$

Divide \left(-7-5i\right)x+\left(4+8i\right) by 2-i.

$$y=\left(-\frac{9}{5}-\frac{17}{5}i\right)x+4i$$