Solve x=y/-1+sqrt{2}

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

Solve for y

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

Steps for Solving Linear Equation

Rationalize the denominator of \frac{y}{-1+\sqrt{2}} by multiplying numerator and denominator by -1-\sqrt{2}.

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

Consider \left(-1+\sqrt{2}\right)\left(-1-\sqrt{2}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Square -1. Square \sqrt{2}.

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

Subtract 2 from 1 to get -1.

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

Anything divided by -1 gives its opposite.

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

Use the distributive property to multiply y by -1-\sqrt{2}.

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

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

$$x=y+y\sqrt{2}$$

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

$$y+y\sqrt{2}=x$$

Combine all terms containing y.

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

The equation is in standard form.

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

Divide both sides by 1+\sqrt{2}.

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

Dividing by 1+\sqrt{2} undoes the multiplication by 1+\sqrt{2}.

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

Divide x by 1+\sqrt{2}.

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

Solve for x

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