Solve 2xleft(4+2sqrt{3}right)-2sqrt{3}left(3sqrt{3}-xright)=sqrt{3}left(5x+3sqrt{3}right)-2left(9-4right)

$$2x \left( 4+2 \sqrt{ 3 } \right) -2 \sqrt{ 3 } \left( 3 \sqrt{ 3 } -x \right) = \sqrt{ 3 } \left( 5x+3 \sqrt{ 3 } \right) -2 \left( 9-4 \right)$$

Solve for x

$$x = \frac{136 - 17 \sqrt{3}}{61} \approx 1.746805513$$

Steps for Solving Linear Equation

Use the distributive property to multiply 2x by 4+2\sqrt{3}.

$$8x+4\sqrt{3}x-2\sqrt{3}\left(3\sqrt{3}-x\right)=\sqrt{3}\left(5x+3\sqrt{3}\right)-2\left(9-4\right)$$

Use the distributive property to multiply \sqrt{3} by 5x+3\sqrt{3}.

$$8x+4\sqrt{3}x-2\sqrt{3}\left(3\sqrt{3}-x\right)=5\sqrt{3}x+3\left(\sqrt{3}\right)^{2}-2\left(9-4\right)$$

The square of \sqrt{3} is 3.

$$8x+4\sqrt{3}x-2\sqrt{3}\left(3\sqrt{3}-x\right)=5\sqrt{3}x+3\times 3-2\left(9-4\right)$$

Multiply 3 and 3 to get 9.

$$8x+4\sqrt{3}x-2\sqrt{3}\left(3\sqrt{3}-x\right)=5\sqrt{3}x+9-2\left(9-4\right)$$

Subtract 4 from 9 to get 5.

$$8x+4\sqrt{3}x-2\sqrt{3}\left(3\sqrt{3}-x\right)=5\sqrt{3}x+9-2\times 5$$

Multiply 2 and 5 to get 10.

$$8x+4\sqrt{3}x-2\sqrt{3}\left(3\sqrt{3}-x\right)=5\sqrt{3}x+9-10$$

Subtract 10 from 9 to get -1.

$$8x+4\sqrt{3}x-2\sqrt{3}\left(3\sqrt{3}-x\right)=5\sqrt{3}x-1$$

Subtract 5\sqrt{3}x from both sides.

$$8x+4\sqrt{3}x-2\sqrt{3}\left(3\sqrt{3}-x\right)-5\sqrt{3}x=-1$$

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

$$8x+4\sqrt{3}x-6\left(\sqrt{3}\right)^{2}+2\sqrt{3}x-5\sqrt{3}x=-1$$

The square of \sqrt{3} is 3.

$$8x+4\sqrt{3}x-6\times 3+2\sqrt{3}x-5\sqrt{3}x=-1$$

Multiply -6 and 3 to get -18.

$$8x+4\sqrt{3}x-18+2\sqrt{3}x-5\sqrt{3}x=-1$$

Combine 4\sqrt{3}x and 2\sqrt{3}x to get 6\sqrt{3}x.

$$8x+6\sqrt{3}x-18-5\sqrt{3}x=-1$$

Combine 6\sqrt{3}x and -5\sqrt{3}x to get \sqrt{3}x.

$$8x+\sqrt{3}x-18=-1$$

Add 18 to both sides.

$$8x+\sqrt{3}x=-1+18$$

Add -1 and 18 to get 17.

$$8x+\sqrt{3}x=17$$

Combine all terms containing x.

$$\left(8+\sqrt{3}\right)x=17$$

The equation is in standard form.

$$\left(\sqrt{3}+8\right)x=17$$

Divide both sides by 8+\sqrt{3}.

$$\frac{\left(\sqrt{3}+8\right)x}{\sqrt{3}+8}=\frac{17}{\sqrt{3}+8}$$

Dividing by 8+\sqrt{3} undoes the multiplication by 8+\sqrt{3}.

$$x=\frac{17}{\sqrt{3}+8}$$

Divide 17 by 8+\sqrt{3}.

$$x=\frac{136-17\sqrt{3}}{61}$$