Solve xsqrt{3+2sqrt{2}}-ysqrt{17-12sqrt{2}}=sqrt{18-8sqrt{2}}

$$x \sqrt { 3 + 2 \sqrt { 2 } } - y \sqrt { 17 - 12 \sqrt { 2 } } = \sqrt { 18 - 8 \sqrt { 2 } }$$

Solve for x

$$x=5\sqrt{2}y-7y+5\sqrt{2}-6$$

Steps for Solving Linear Equation

Add y\sqrt{17-12\sqrt{2}} to both sides.

$$x\sqrt{3+2\sqrt{2}}=\sqrt{18-8\sqrt{2}}+y\sqrt{17-12\sqrt{2}}$$

The equation is in standard form.

$$\sqrt{2\sqrt{2}+3}x=\sqrt{17-12\sqrt{2}}y+\sqrt{18-8\sqrt{2}}$$

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

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

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

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

Divide 4-\sqrt{2}+y\left(3-2\sqrt{2}\right) by \sqrt{3+2\sqrt{2}}.

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

Solve for y

$$y=5\sqrt{2}\left(x-1\right)+7x-8$$

Steps for Solving Linear Equation

Subtract x\sqrt{3+2\sqrt{2}} from both sides.

$$-y\sqrt{17-12\sqrt{2}}=\sqrt{18-8\sqrt{2}}-x\sqrt{3+2\sqrt{2}}$$

The equation is in standard form.

$$\left(-\sqrt{17-12\sqrt{2}}\right)y=-\sqrt{2\sqrt{2}+3}x+\sqrt{18-8\sqrt{2}}$$

Divide both sides by -\sqrt{17-12\sqrt{2}}.

$$\frac{\left(-\sqrt{17-12\sqrt{2}}\right)y}{-\sqrt{17-12\sqrt{2}}}=\frac{-\left(\sqrt{2}+1\right)x+4-\sqrt{2}}{-\sqrt{17-12\sqrt{2}}}$$

Dividing by -\sqrt{17-12\sqrt{2}} undoes the multiplication by -\sqrt{17-12\sqrt{2}}.

$$y=\frac{-\left(\sqrt{2}+1\right)x+4-\sqrt{2}}{-\sqrt{17-12\sqrt{2}}}$$

Divide 4-\sqrt{2}-x\left(\sqrt{2}+1\right) by -\sqrt{17-12\sqrt{2}}.

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