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

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

Solve for x

$$x=\frac{2\sqrt{5}}{7}\approx 0.638876565$$

Steps for Solving Linear Equation

Rationalize the denominator of \frac{x}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

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

The square of \sqrt{5} is 5.

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

Divide \frac{2\sqrt{5}}{5}-x by \frac{2\sqrt{5}}{5} by multiplying \frac{2\sqrt{5}}{5}-x by the reciprocal of \frac{2\sqrt{5}}{5}.

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

Rationalize the denominator of \frac{\left(\frac{2\sqrt{5}}{5}-x\right)\times 5}{2\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

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

The square of \sqrt{5} is 5.

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

Multiply 2 and 5 to get 10.

$$\frac{x\sqrt{5}}{5}=\frac{\left(\frac{2\sqrt{5}}{5}-x\right)\times 5\sqrt{5}}{10}$$

Divide \left(\frac{2\sqrt{5}}{5}-x\right)\times 5\sqrt{5} by 10 to get \left(\frac{2\sqrt{5}}{5}-x\right)\times \frac{1}{2}\sqrt{5}.

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

Use the distributive property to multiply \frac{2\sqrt{5}}{5}-x by \frac{1}{2}.

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

Multiply \frac{2\sqrt{5}}{5} times \frac{1}{2} by multiplying numerator times numerator and denominator times denominator.

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

Cancel out 2 in both numerator and denominator.

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

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

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

Use the distributive property to multiply \frac{\sqrt{5}}{5}-\frac{1}{2}x by \sqrt{5}.

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

Express \frac{\sqrt{5}}{5}\sqrt{5} as a single fraction.

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

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

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

Divide 5 by 5 to get 1.

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

Add \frac{1}{2}x\sqrt{5} to both sides.

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

Combine \frac{x\sqrt{5}}{5} and \frac{1}{2}x\sqrt{5} to get \frac{7}{10}x\sqrt{5}.

$$\frac{7}{10}x\sqrt{5}=1$$

The equation is in standard form.

$$\frac{7\sqrt{5}}{10}x=1$$

Divide both sides by \frac{7}{10}\sqrt{5}.

$$\frac{10\times \frac{7\sqrt{5}}{10}x}{7\sqrt{5}}=\frac{10}{7\sqrt{5}}$$

Dividing by \frac{7}{10}\sqrt{5} undoes the multiplication by \frac{7}{10}\sqrt{5}.

$$x=\frac{10}{7\sqrt{5}}$$

Divide 1 by \frac{7}{10}\sqrt{5}.

$$x=\frac{2\sqrt{5}}{7}$$