Solve x/sqrt{3/3}-x/2+sqrt{3}=2

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

Solve for x

$$x = \frac{\sqrt{3} + 1}{2} \approx 1.366025404$$

Steps for Solving Linear Equation

Divide x by \frac{\sqrt{3}}{3} by multiplying x by the reciprocal of \frac{\sqrt{3}}{3}.

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

Rationalize the denominator of \frac{x\times 3}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{x\times 3\sqrt{3}}{\left(\sqrt{3}\right)^{2}}-\frac{x}{2+\sqrt{3}}=2$$

The square of \sqrt{3} is 3.

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

Rationalize the denominator of \frac{x}{2+\sqrt{3}} by multiplying numerator and denominator by 2-\sqrt{3}.

$$\frac{x\times 3\sqrt{3}}{3}-\frac{x\left(2-\sqrt{3}\right)}{\left(2+\sqrt{3}\right)\left(2-\sqrt{3}\right)}=2$$

Consider \left(2+\sqrt{3}\right)\left(2-\sqrt{3}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{x\times 3\sqrt{3}}{3}-\frac{x\left(2-\sqrt{3}\right)}{2^{2}-\left(\sqrt{3}\right)^{2}}=2$$

Square 2. Square \sqrt{3}.

$$\frac{x\times 3\sqrt{3}}{3}-\frac{x\left(2-\sqrt{3}\right)}{4-3}=2$$

Subtract 3 from 4 to get 1.

$$\frac{x\times 3\sqrt{3}}{3}-\frac{x\left(2-\sqrt{3}\right)}{1}=2$$

Anything divided by one gives itself.

$$\frac{x\times 3\sqrt{3}}{3}-x\left(2-\sqrt{3}\right)=2$$

Use the distributive property to multiply x by 2-\sqrt{3}.

$$\frac{x\times 3\sqrt{3}}{3}-\left(2x-x\sqrt{3}\right)=2$$

To find the opposite of 2x-x\sqrt{3}, find the opposite of each term.

$$\frac{x\times 3\sqrt{3}}{3}-2x-\left(-x\sqrt{3}\right)=2$$

The opposite of -x\sqrt{3} is x\sqrt{3}.

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

Cancel out 3 and 3.

$$x\sqrt{3}-2x+x\sqrt{3}=2$$

Combine x\sqrt{3} and x\sqrt{3} to get 2x\sqrt{3}.

$$2x\sqrt{3}-2x=2$$

Combine all terms containing x.

$$\left(2\sqrt{3}-2\right)x=2$$

Divide both sides by 2\sqrt{3}-2.

$$\frac{\left(2\sqrt{3}-2\right)x}{2\sqrt{3}-2}=\frac{2}{2\sqrt{3}-2}$$

Dividing by 2\sqrt{3}-2 undoes the multiplication by 2\sqrt{3}-2.

$$x=\frac{2}{2\sqrt{3}-2}$$

Divide 2 by 2\sqrt{3}-2.

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