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

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

Solve for x

$$x=\left(\frac{7}{29}-\frac{3}{29}i\right)y+\left(\frac{5}{29}+\frac{2}{29}i\right)$$

Steps for Solving Linear Equation

Add \left(1+i\right)y to both sides.

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

The equation is in standard form.

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

Divide both sides by 2+5i.

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

Dividing by 2+5i undoes the multiplication by 2+5i.

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

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

$$x=\left(\frac{7}{29}-\frac{3}{29}i\right)y+\left(\frac{5}{29}+\frac{2}{29}i\right)$$

Solve for y

$$y=\left(\frac{7}{2}+\frac{3}{2}i\right)x+\left(-\frac{1}{2}-\frac{1}{2}i\right)$$

Steps for Solving Linear Equation

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

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

Subtract \left(2+5i\right)x from both sides.

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

The equation is in standard form.

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

Divide both sides by -1-i.

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

Dividing by -1-i undoes the multiplication by -1-i.

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

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

$$y=\left(\frac{7}{2}+\frac{3}{2}i\right)x+\left(-\frac{1}{2}-\frac{1}{2}i\right)$$