Solve r/10=10-r/10sqrt{2}

$$\frac { r } { 10 } = \frac { 10 - r } { 10 \sqrt { 2 } }$$

Solve for r

$$r=10\left(\sqrt{2}-1\right)\approx 4.142135624$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 10.

$$r=\frac{1}{2}\times 2^{\frac{1}{2}}\left(10-r\right)$$

Use the distributive property to multiply \frac{1}{2}\times 2^{\frac{1}{2}} by 10-r.

$$r=\frac{1}{2}\times 2^{\frac{1}{2}}\times 10+\frac{1}{2}\times 2^{\frac{1}{2}}\left(-1\right)r$$

Multiply \frac{1}{2} and 10 to get \frac{10}{2}.

$$r=\frac{10}{2}\times 2^{\frac{1}{2}}+\frac{1}{2}\times 2^{\frac{1}{2}}\left(-1\right)r$$

Divide 10 by 2 to get 5.

$$r=5\times 2^{\frac{1}{2}}+\frac{1}{2}\times 2^{\frac{1}{2}}\left(-1\right)r$$

Multiply \frac{1}{2} and -1 to get -\frac{1}{2}.

$$r=5\times 2^{\frac{1}{2}}-\frac{1}{2}\times 2^{\frac{1}{2}}r$$

Add \frac{1}{2}\times 2^{\frac{1}{2}}r to both sides.

$$r+\frac{1}{2}\times 2^{\frac{1}{2}}r=5\times 2^{\frac{1}{2}}$$

Reorder the terms.

$$\frac{1}{2}\sqrt{2}r+r=5\sqrt{2}$$

Combine all terms containing r.

$$\left(\frac{1}{2}\sqrt{2}+1\right)r=5\sqrt{2}$$

The equation is in standard form.

$$\left(\frac{\sqrt{2}}{2}+1\right)r=5\sqrt{2}$$

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

$$\frac{\left(\frac{\sqrt{2}}{2}+1\right)r}{\frac{\sqrt{2}}{2}+1}=\frac{5\sqrt{2}}{\frac{\sqrt{2}}{2}+1}$$

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

$$r=\frac{5\sqrt{2}}{\frac{\sqrt{2}}{2}+1}$$

Divide 5\sqrt{2} by \frac{1}{2}\sqrt{2}+1.

$$r=10\sqrt{2}-10$$