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

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

Solve for x

$$x=\sqrt{5}\approx 2.236067977$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 2.

$$\left(-\frac{1}{2}\times 5^{\frac{1}{2}}-\frac{1}{2}\right)\left(x-2\right)+\sqrt{5}x=\left(\frac{1}{2}\times 5^{\frac{1}{2}}-\frac{1}{2}\right)\left(\sqrt{5}-2\right)+2\sqrt{5}$$

Apply the distributive property by multiplying each term of -\frac{1}{2}\times 5^{\frac{1}{2}}-\frac{1}{2} by each term of x-2.

$$-\frac{1}{2}\times 5^{\frac{1}{2}}x-\frac{1}{2}\times 5^{\frac{1}{2}}\left(-2\right)-\frac{1}{2}x-\frac{1}{2}\left(-2\right)+\sqrt{5}x=\left(\frac{1}{2}\times 5^{\frac{1}{2}}-\frac{1}{2}\right)\left(\sqrt{5}-2\right)+2\sqrt{5}$$

Express -\frac{1}{2}\left(-2\right) as a single fraction.

$$-\frac{1}{2}\times 5^{\frac{1}{2}}x+\frac{-\left(-2\right)}{2}\times 5^{\frac{1}{2}}-\frac{1}{2}x-\frac{1}{2}\left(-2\right)+\sqrt{5}x=\left(\frac{1}{2}\times 5^{\frac{1}{2}}-\frac{1}{2}\right)\left(\sqrt{5}-2\right)+2\sqrt{5}$$

Multiply -1 and -2 to get 2.

$$-\frac{1}{2}\times 5^{\frac{1}{2}}x+\frac{2}{2}\times 5^{\frac{1}{2}}-\frac{1}{2}x-\frac{1}{2}\left(-2\right)+\sqrt{5}x=\left(\frac{1}{2}\times 5^{\frac{1}{2}}-\frac{1}{2}\right)\left(\sqrt{5}-2\right)+2\sqrt{5}$$

Divide 2 by 2 to get 1.

$$-\frac{1}{2}\times 5^{\frac{1}{2}}x+1\times 5^{\frac{1}{2}}-\frac{1}{2}x-\frac{1}{2}\left(-2\right)+\sqrt{5}x=\left(\frac{1}{2}\times 5^{\frac{1}{2}}-\frac{1}{2}\right)\left(\sqrt{5}-2\right)+2\sqrt{5}$$

Express -\frac{1}{2}\left(-2\right) as a single fraction.

$$-\frac{1}{2}\times 5^{\frac{1}{2}}x+1\times 5^{\frac{1}{2}}-\frac{1}{2}x+\frac{-\left(-2\right)}{2}+\sqrt{5}x=\left(\frac{1}{2}\times 5^{\frac{1}{2}}-\frac{1}{2}\right)\left(\sqrt{5}-2\right)+2\sqrt{5}$$

Multiply -1 and -2 to get 2.

$$-\frac{1}{2}\times 5^{\frac{1}{2}}x+1\times 5^{\frac{1}{2}}-\frac{1}{2}x+\frac{2}{2}+\sqrt{5}x=\left(\frac{1}{2}\times 5^{\frac{1}{2}}-\frac{1}{2}\right)\left(\sqrt{5}-2\right)+2\sqrt{5}$$

Divide 2 by 2 to get 1.

$$-\frac{1}{2}\times 5^{\frac{1}{2}}x+1\times 5^{\frac{1}{2}}-\frac{1}{2}x+1+\sqrt{5}x=\left(\frac{1}{2}\times 5^{\frac{1}{2}}-\frac{1}{2}\right)\left(\sqrt{5}-2\right)+2\sqrt{5}$$

Apply the distributive property by multiplying each term of \frac{1}{2}\times 5^{\frac{1}{2}}-\frac{1}{2} by each term of \sqrt{5}-2.

$$-\frac{1}{2}\times 5^{\frac{1}{2}}x+1\times 5^{\frac{1}{2}}-\frac{1}{2}x+1+\sqrt{5}x=\frac{1}{2}\times 5^{\frac{1}{2}}\sqrt{5}+\frac{1}{2}\times 5^{\frac{1}{2}}\left(-2\right)-\frac{1}{2}\sqrt{5}-\frac{1}{2}\left(-2\right)+2\sqrt{5}$$

Multiply \frac{1}{2} and -2 to get \frac{-2}{2}.

$$-\frac{1}{2}\times 5^{\frac{1}{2}}x+1\times 5^{\frac{1}{2}}-\frac{1}{2}x+1+\sqrt{5}x=\frac{1}{2}\times 5^{\frac{1}{2}}\sqrt{5}+\frac{-2}{2}\times 5^{\frac{1}{2}}-\frac{1}{2}\sqrt{5}-\frac{1}{2}\left(-2\right)+2\sqrt{5}$$

Divide -2 by 2 to get -1.

$$-\frac{1}{2}\times 5^{\frac{1}{2}}x+1\times 5^{\frac{1}{2}}-\frac{1}{2}x+1+\sqrt{5}x=\frac{1}{2}\times 5^{\frac{1}{2}}\sqrt{5}-5^{\frac{1}{2}}-\frac{1}{2}\sqrt{5}-\frac{1}{2}\left(-2\right)+2\sqrt{5}$$

Express -\frac{1}{2}\left(-2\right) as a single fraction.

$$-\frac{1}{2}\times 5^{\frac{1}{2}}x+1\times 5^{\frac{1}{2}}-\frac{1}{2}x+1+\sqrt{5}x=\frac{1}{2}\times 5^{\frac{1}{2}}\sqrt{5}-5^{\frac{1}{2}}-\frac{1}{2}\sqrt{5}+\frac{-\left(-2\right)}{2}+2\sqrt{5}$$

Multiply -1 and -2 to get 2.

$$-\frac{1}{2}\times 5^{\frac{1}{2}}x+1\times 5^{\frac{1}{2}}-\frac{1}{2}x+1+\sqrt{5}x=\frac{1}{2}\times 5^{\frac{1}{2}}\sqrt{5}-5^{\frac{1}{2}}-\frac{1}{2}\sqrt{5}+\frac{2}{2}+2\sqrt{5}$$

Divide 2 by 2 to get 1.

$$-\frac{1}{2}\times 5^{\frac{1}{2}}x+1\times 5^{\frac{1}{2}}-\frac{1}{2}x+1+\sqrt{5}x=\frac{1}{2}\times 5^{\frac{1}{2}}\sqrt{5}-5^{\frac{1}{2}}-\frac{1}{2}\sqrt{5}+1+2\sqrt{5}$$

Combine -\frac{1}{2}\sqrt{5} and 2\sqrt{5} to get \frac{3}{2}\sqrt{5}.

$$-\frac{1}{2}\times 5^{\frac{1}{2}}x+1\times 5^{\frac{1}{2}}-\frac{1}{2}x+1+\sqrt{5}x=\frac{1}{2}\times 5^{\frac{1}{2}}\sqrt{5}-5^{\frac{1}{2}}+\frac{3}{2}\sqrt{5}+1$$

Subtract 1\times 5^{\frac{1}{2}} from both sides.

$$-\frac{1}{2}\times 5^{\frac{1}{2}}x-\frac{1}{2}x+1+\sqrt{5}x=\frac{1}{2}\times 5^{\frac{1}{2}}\sqrt{5}-5^{\frac{1}{2}}+\frac{3}{2}\sqrt{5}+1-1\times 5^{\frac{1}{2}}$$

Combine -5^{\frac{1}{2}} and -5^{\frac{1}{2}} to get -2\times 5^{\frac{1}{2}}.

$$-\frac{1}{2}\times 5^{\frac{1}{2}}x-\frac{1}{2}x+1+\sqrt{5}x=\frac{1}{2}\times 5^{\frac{1}{2}}\sqrt{5}-2\times 5^{\frac{1}{2}}+\frac{3}{2}\sqrt{5}+1$$

Subtract 1 from both sides.

$$-\frac{1}{2}\times 5^{\frac{1}{2}}x-\frac{1}{2}x+\sqrt{5}x=\frac{1}{2}\times 5^{\frac{1}{2}}\sqrt{5}-2\times 5^{\frac{1}{2}}+\frac{3}{2}\sqrt{5}+1-1$$

Subtract 1 from 1 to get 0.

$$-\frac{1}{2}\times 5^{\frac{1}{2}}x-\frac{1}{2}x+\sqrt{5}x=\frac{1}{2}\times 5^{\frac{1}{2}}\sqrt{5}-2\times 5^{\frac{1}{2}}+\frac{3}{2}\sqrt{5}$$

Reorder the terms.

$$\sqrt{5}x-\frac{1}{2}\sqrt{5}x-\frac{1}{2}x=\frac{1}{2}\sqrt{5}\sqrt{5}-2\sqrt{5}+\frac{3}{2}\sqrt{5}$$

Multiply \sqrt{5} and \sqrt{5} to get 5.

$$\sqrt{5}x-\frac{1}{2}\sqrt{5}x-\frac{1}{2}x=\frac{1}{2}\times 5-2\sqrt{5}+\frac{3}{2}\sqrt{5}$$

Combine \sqrt{5}x and -\frac{1}{2}\sqrt{5}x to get \frac{1}{2}\sqrt{5}x.

$$\frac{1}{2}\sqrt{5}x-\frac{1}{2}x=\frac{1}{2}\times 5-2\sqrt{5}+\frac{3}{2}\sqrt{5}$$

Multiply \frac{1}{2} and 5 to get \frac{5}{2}.

$$\frac{1}{2}\sqrt{5}x-\frac{1}{2}x=\frac{5}{2}-2\sqrt{5}+\frac{3}{2}\sqrt{5}$$

Combine -2\sqrt{5} and \frac{3}{2}\sqrt{5} to get -\frac{1}{2}\sqrt{5}.

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

Combine all terms containing x.

$$\left(\frac{1}{2}\sqrt{5}-\frac{1}{2}\right)x=\frac{5}{2}-\frac{1}{2}\sqrt{5}$$

The equation is in standard form.

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

Divide both sides by \frac{1}{2}\sqrt{5}-\frac{1}{2}.

$$\frac{2\times \frac{\sqrt{5}-1}{2}x}{\sqrt{5}-1}=\frac{5-\sqrt{5}}{2\times \frac{\sqrt{5}-1}{2}}$$

Dividing by \frac{1}{2}\sqrt{5}-\frac{1}{2} undoes the multiplication by \frac{1}{2}\sqrt{5}-\frac{1}{2}.

$$x=\frac{5-\sqrt{5}}{2\times \frac{\sqrt{5}-1}{2}}$$

Divide \frac{5-\sqrt{5}}{2} by \frac{1}{2}\sqrt{5}-\frac{1}{2}.

$$x=\sqrt{5}$$