Solve y/sqrt{3+frac{sqrt{6}}{2}}=x/1+3sqrt{2/2}

$$\frac { y } { \sqrt { 3 } + \frac { \sqrt { 6 } } { 2 } } = \frac { x } { 1 + \frac { 3 \sqrt { 2 } } { 2 } }$$

Solve for x

$$x=-\frac{\left(\sqrt{3}-2\sqrt{6}\right)y}{3}$$

Steps for Solving Linear Equation

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

$$\frac{x}{1+\frac{3\sqrt{2}}{2}}=\frac{y}{\sqrt{3}+\frac{\sqrt{6}}{2}}$$

The equation is in standard form.

$$\frac{1}{\frac{3\sqrt{2}}{2}+1}x=\frac{y}{\frac{\sqrt{6}}{2}+\sqrt{3}}$$

Divide both sides by \left(1+\frac{3}{2}\sqrt{2}\right)^{-1}.

$$\frac{\frac{1}{\frac{3\sqrt{2}}{2}+1}x\left(\frac{3\sqrt{2}}{2}+1\right)}{1}=\frac{\left(-\frac{\left(\sqrt{6}-2\sqrt{3}\right)y}{3}\right)\left(\frac{3\sqrt{2}}{2}+1\right)}{1}$$

Dividing by \left(1+\frac{3}{2}\sqrt{2}\right)^{-1} undoes the multiplication by \left(1+\frac{3}{2}\sqrt{2}\right)^{-1}.

$$x=\frac{\left(-\frac{\left(\sqrt{6}-2\sqrt{3}\right)y}{3}\right)\left(\frac{3\sqrt{2}}{2}+1\right)}{1}$$

Divide -\frac{y\left(\sqrt{6}-2\sqrt{3}\right)}{3} by \left(1+\frac{3}{2}\sqrt{2}\right)^{-1}.

$$x=-\frac{\left(2\sqrt{3}-4\sqrt{6}\right)y}{6}$$

Solve for y

$$y=\frac{\left(\sqrt{3}+2\sqrt{6}\right)x}{7}$$

Steps for Solving Linear Equation

The equation is in standard form.

$$\frac{1}{\frac{\sqrt{6}}{2}+\sqrt{3}}y=\frac{x}{\frac{3\sqrt{2}}{2}+1}$$

Divide both sides by \left(\sqrt{3}+\frac{1}{2}\sqrt{6}\right)^{-1}.

$$\frac{\frac{1}{\frac{\sqrt{6}}{2}+\sqrt{3}}y\left(\frac{\sqrt{6}}{2}+\sqrt{3}\right)}{1}=\frac{3\sqrt{2}x-2x}{7\times \frac{1}{\frac{\sqrt{6}}{2}+\sqrt{3}}}$$

Dividing by \left(\sqrt{3}+\frac{1}{2}\sqrt{6}\right)^{-1} undoes the multiplication by \left(\sqrt{3}+\frac{1}{2}\sqrt{6}\right)^{-1}.

$$y=\frac{3\sqrt{2}x-2x}{7\times \frac{1}{\frac{\sqrt{6}}{2}+\sqrt{3}}}$$

Divide \frac{-2x+3x\sqrt{2}}{7} by \left(\sqrt{3}+\frac{1}{2}\sqrt{6}\right)^{-1}.

$$y=\frac{\left(2\sqrt{3}+4\sqrt{6}\right)x}{14}$$