Solve frac{x+2ysqrt{-1}}{2-sqrt{-1}}+frac{2x+ysqrt{-1}}{2+sqrt{-1}}=23/5

$$\frac{ x+2y \sqrt{ -1 } }{ 2- \sqrt{ -1 } } + \frac{ 2x+y \sqrt{ -1 } }{ 2+ \sqrt{ -1 } } = \frac{ 23 }{ 5 }$$

Solve for y

$$y=\left(\frac{12}{37}+\frac{35}{37}i\right)x+\left(-\frac{23}{37}-\frac{138}{37}i\right)$$

Steps for Solving Linear Equation

Calculate the square root of -1 and get i.

$$\frac{x+2yi}{2-\sqrt{-1}}+\frac{2x+y\sqrt{-1}}{2+\sqrt{-1}}=\frac{23}{5}$$

Multiply 2 and i to get 2i.

$$\frac{x+2iy}{2-\sqrt{-1}}+\frac{2x+y\sqrt{-1}}{2+\sqrt{-1}}=\frac{23}{5}$$

Calculate the square root of -1 and get i.

$$\frac{x+2iy}{2-i}+\frac{2x+y\sqrt{-1}}{2+\sqrt{-1}}=\frac{23}{5}$$

Calculate the square root of -1 and get i.

$$\frac{x+2iy}{2-i}+\frac{2x+yi}{2+\sqrt{-1}}=\frac{23}{5}$$

Calculate the square root of -1 and get i.

$$\frac{x+2iy}{2-i}+\frac{2x+yi}{2+i}=\frac{23}{5}$$

Divide each term of x+2iy by 2-i to get \frac{x}{2-i}+\frac{2iy}{2-i}.

$$\frac{x}{2-i}+\frac{2iy}{2-i}+\frac{2x+yi}{2+i}=\frac{23}{5}$$

Divide 2iy by 2-i to get \left(-\frac{2}{5}+\frac{4}{5}i\right)y.

$$\frac{x}{2-i}+\left(-\frac{2}{5}+\frac{4}{5}i\right)y+\frac{2x+yi}{2+i}=\frac{23}{5}$$

Divide each term of 2x+yi by 2+i to get \frac{2x}{2+i}+\frac{yi}{2+i}.

$$\frac{x}{2-i}+\left(-\frac{2}{5}+\frac{4}{5}i\right)y+\frac{2x}{2+i}+\frac{yi}{2+i}=\frac{23}{5}$$

Divide 2x by 2+i to get \left(\frac{4}{5}-\frac{2}{5}i\right)x.

$$\frac{x}{2-i}+\left(-\frac{2}{5}+\frac{4}{5}i\right)y+\left(\frac{4}{5}-\frac{2}{5}i\right)x+\frac{yi}{2+i}=\frac{23}{5}$$

Divide yi by 2+i to get y\left(\frac{1}{5}+\frac{2}{5}i\right).

$$\frac{x}{2-i}+\left(-\frac{2}{5}+\frac{4}{5}i\right)y+\left(\frac{4}{5}-\frac{2}{5}i\right)x+y\left(\frac{1}{5}+\frac{2}{5}i\right)=\frac{23}{5}$$

Combine \left(-\frac{2}{5}+\frac{4}{5}i\right)y and y\left(\frac{1}{5}+\frac{2}{5}i\right) to get \left(-\frac{1}{5}+\frac{6}{5}i\right)y.

$$\frac{x}{2-i}+\left(-\frac{1}{5}+\frac{6}{5}i\right)y+\left(\frac{4}{5}-\frac{2}{5}i\right)x=\frac{23}{5}$$

Subtract \frac{x}{2-i} from both sides.

$$\left(-\frac{1}{5}+\frac{6}{5}i\right)y+\left(\frac{4}{5}-\frac{2}{5}i\right)x=\frac{23}{5}-\frac{x}{2-i}$$

Subtract \left(\frac{4}{5}-\frac{2}{5}i\right)x from both sides.

$$\left(-\frac{1}{5}+\frac{6}{5}i\right)y=\frac{23}{5}-\frac{x}{2-i}-\left(\frac{4}{5}-\frac{2}{5}i\right)x$$

Multiply -1 and \frac{4}{5}-\frac{2}{5}i to get -\frac{4}{5}+\frac{2}{5}i.

$$\left(-\frac{1}{5}+\frac{6}{5}i\right)y=\frac{23}{5}-\frac{x}{2-i}+\left(-\frac{4}{5}+\frac{2}{5}i\right)x$$

The equation is in standard form.

$$\left(-\frac{1}{5}+\frac{6}{5}i\right)y=\left(-\frac{6}{5}+\frac{1}{5}i\right)x+\frac{23}{5}$$

Divide both sides by -\frac{1}{5}+\frac{6}{5}i.

$$\frac{\left(-\frac{1}{5}+\frac{6}{5}i\right)y}{-\frac{1}{5}+\frac{6}{5}i}=\frac{\left(-\frac{6}{5}+\frac{1}{5}i\right)x+\frac{23}{5}}{-\frac{1}{5}+\frac{6}{5}i}$$

Dividing by -\frac{1}{5}+\frac{6}{5}i undoes the multiplication by -\frac{1}{5}+\frac{6}{5}i.

$$y=\frac{\left(-\frac{6}{5}+\frac{1}{5}i\right)x+\frac{23}{5}}{-\frac{1}{5}+\frac{6}{5}i}$$

Divide \frac{23}{5}+\left(-\frac{6}{5}+\frac{1}{5}i\right)x by -\frac{1}{5}+\frac{6}{5}i.

$$y=\left(\frac{12}{37}+\frac{35}{37}i\right)x+\left(-\frac{23}{37}-\frac{138}{37}i\right)$$

Solve for x

$$x=\left(\frac{12}{37}-\frac{35}{37}i\right)y+\left(\frac{138}{37}+\frac{23}{37}i\right)$$