Solve 2/w=1+i/sqrt{3+i}

$$\frac { 2 } { w } = \frac { 1 + i } { \sqrt { 3 } + i }$$

Solve for w

$$w=-\frac{8}{\sqrt{3}\left(-1-i\right)+\left(-1+i\right)}\approx 2.732050808-0.732050808i$$

Steps for Solving Linear Equation

Variable w cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by w.

$$2=\frac{1}{4}w\left(3^{\frac{1}{2}}-i\right)\left(1+i\right)$$

Multiply \frac{1}{4} times 1+i.

$$2=\left(\frac{1}{4}\times 1+\frac{1}{4}i\right)w\left(3^{\frac{1}{2}}-i\right)$$

Do the multiplications in \frac{1}{4}\times 1+\frac{1}{4}i.

$$2=\left(\frac{1}{4}+\frac{1}{4}i\right)w\left(3^{\frac{1}{2}}-i\right)$$

Use the distributive property to multiply \left(\frac{1}{4}+\frac{1}{4}i\right)w by 3^{\frac{1}{2}}-i.

$$2=\left(\frac{1}{4}+\frac{1}{4}i\right)w\times 3^{\frac{1}{2}}+\left(\frac{1}{4}-\frac{1}{4}i\right)w$$

Swap sides so that all variable terms are on the left hand side.

$$\left(\frac{1}{4}+\frac{1}{4}i\right)w\times 3^{\frac{1}{2}}+\left(\frac{1}{4}-\frac{1}{4}i\right)w=2$$

Reorder the terms.

$$\sqrt{3}\left(\frac{1}{4}+\frac{1}{4}i\right)w+\left(\frac{1}{4}-\frac{1}{4}i\right)w=2$$

Combine all terms containing w.

$$\left(\sqrt{3}\left(\frac{1}{4}+\frac{1}{4}i\right)+\left(\frac{1}{4}-\frac{1}{4}i\right)\right)w=2$$

Divide both sides by \left(\frac{1}{4}+\frac{1}{4}i\right)\sqrt{3}+\left(\frac{1}{4}-\frac{1}{4}i\right).

$$\frac{\left(\sqrt{3}\left(\frac{1}{4}+\frac{1}{4}i\right)+\left(\frac{1}{4}-\frac{1}{4}i\right)\right)w}{\sqrt{3}\left(\frac{1}{4}+\frac{1}{4}i\right)+\left(\frac{1}{4}-\frac{1}{4}i\right)}=\frac{2}{\sqrt{3}\left(\frac{1}{4}+\frac{1}{4}i\right)+\left(\frac{1}{4}-\frac{1}{4}i\right)}$$

Dividing by \left(\frac{1}{4}+\frac{1}{4}i\right)\sqrt{3}+\left(\frac{1}{4}-\frac{1}{4}i\right) undoes the multiplication by \left(\frac{1}{4}+\frac{1}{4}i\right)\sqrt{3}+\left(\frac{1}{4}-\frac{1}{4}i\right).

$$w=\frac{2}{\sqrt{3}\left(\frac{1}{4}+\frac{1}{4}i\right)+\left(\frac{1}{4}-\frac{1}{4}i\right)}$$

Divide 2 by \left(\frac{1}{4}+\frac{1}{4}i\right)\sqrt{3}+\left(\frac{1}{4}-\frac{1}{4}i\right).

$$w=\frac{8}{\sqrt{3}\left(1+i\right)+\left(1-i\right)}$$