Solve 9(sqrt{7}+sqrt{3})=sqrt{12}x+sqrt{14}x+1/2x-x

$$9( \sqrt{ 7 } + \sqrt{ 3 } )= \sqrt{ 12 } x+ \sqrt{ 14 } x+ \frac{ 1 }{ 2 } x-x$$

Solve for x

$$x = \frac{18 {(\sqrt{3} + \sqrt{7})}}{1 - 2 \sqrt{14} - 4 \sqrt{3}} \approx 5.875579343$$

Steps for Solving Linear Equation

Use the distributive property to multiply 9 by \sqrt{7}+\sqrt{3}.

$$9\sqrt{7}+9\sqrt{3}=\sqrt{12}x+\sqrt{14}x+\frac{1}{2}x-x$$

Factor 12=2^{2}\times 3. Rewrite the square root of the product \sqrt{2^{2}\times 3} as the product of square roots \sqrt{2^{2}}\sqrt{3}. Take the square root of 2^{2}.

$$9\sqrt{7}+9\sqrt{3}=2\sqrt{3}x+\sqrt{14}x+\frac{1}{2}x-x$$

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

$$9\sqrt{7}+9\sqrt{3}=2\sqrt{3}x+\sqrt{14}x-\frac{1}{2}x$$

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

$$2\sqrt{3}x+\sqrt{14}x-\frac{1}{2}x=9\sqrt{7}+9\sqrt{3}$$

Combine all terms containing x.

$$\left(2\sqrt{3}+\sqrt{14}-\frac{1}{2}\right)x=9\sqrt{7}+9\sqrt{3}$$

The equation is in standard form.

$$\left(\sqrt{14}+2\sqrt{3}-\frac{1}{2}\right)x=9\sqrt{3}+9\sqrt{7}$$

Divide both sides by 2\sqrt{3}+\sqrt{14}-\frac{1}{2}.

$$\frac{\left(\sqrt{14}+2\sqrt{3}-\frac{1}{2}\right)x}{\sqrt{14}+2\sqrt{3}-\frac{1}{2}}=\frac{9\sqrt{3}+9\sqrt{7}}{\sqrt{14}+2\sqrt{3}-\frac{1}{2}}$$

Dividing by 2\sqrt{3}+\sqrt{14}-\frac{1}{2} undoes the multiplication by 2\sqrt{3}+\sqrt{14}-\frac{1}{2}.

$$x=\frac{9\sqrt{3}+9\sqrt{7}}{\sqrt{14}+2\sqrt{3}-\frac{1}{2}}$$

Divide 9\sqrt{7}+9\sqrt{3} by 2\sqrt{3}+\sqrt{14}-\frac{1}{2}.

$$x=\frac{18\left(\sqrt{3}+\sqrt{7}\right)}{2\sqrt{14}+4\sqrt{3}-1}$$