Solve (3-sqrt{2})(k/sqrt{2}-3)=k

$$( 3 - \sqrt { 2 } ) ( \frac { k } { \sqrt { 2 } } - 3 ) = k$$

Solve for k

$$k=15\sqrt{2}+18\approx 39.213203436$$

Steps for Solving Linear Equation

Rationalize the denominator of \frac{k}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\left(3-\sqrt{2}\right)\left(\frac{k\sqrt{2}}{\left(\sqrt{2}\right)^{2}}-3\right)=k$$

The square of \sqrt{2} is 2.

$$\left(3-\sqrt{2}\right)\left(\frac{k\sqrt{2}}{2}-3\right)=k$$

Apply the distributive property by multiplying each term of 3-\sqrt{2} by each term of \frac{k\sqrt{2}}{2}-3.

$$3\times \frac{k\sqrt{2}}{2}-9-\sqrt{2}\times \frac{k\sqrt{2}}{2}+3\sqrt{2}=k$$

Express 3\times \frac{k\sqrt{2}}{2} as a single fraction.

$$\frac{3k\sqrt{2}}{2}-9-\sqrt{2}\times \frac{k\sqrt{2}}{2}+3\sqrt{2}=k$$

Express \sqrt{2}\times \frac{k\sqrt{2}}{2} as a single fraction.

$$\frac{3k\sqrt{2}}{2}-9-\frac{\sqrt{2}k\sqrt{2}}{2}+3\sqrt{2}=k$$

Since \frac{3k\sqrt{2}}{2} and \frac{\sqrt{2}k\sqrt{2}}{2} have the same denominator, subtract them by subtracting their numerators.

$$\frac{3k\sqrt{2}-\sqrt{2}k\sqrt{2}}{2}-9+3\sqrt{2}=k$$

Do the multiplications in 3k\sqrt{2}-\sqrt{2}k\sqrt{2}.

$$\frac{3k\sqrt{2}-2k}{2}-9+3\sqrt{2}=k$$

Subtract k from both sides.

$$\frac{3k\sqrt{2}-2k}{2}-9+3\sqrt{2}-k=0$$

Add 9 to both sides. Anything plus zero gives itself.

$$\frac{3k\sqrt{2}-2k}{2}+3\sqrt{2}-k=9$$

Subtract 3\sqrt{2} from both sides.

$$\frac{3k\sqrt{2}-2k}{2}-k=9-3\sqrt{2}$$

Multiply both sides of the equation by 2.

$$3k\sqrt{2}-2k-2k=18-6\sqrt{2}$$

Reorder the terms.

$$3\sqrt{2}k-2k-2k=-6\sqrt{2}+18$$

Combine -2k and -2k to get -4k.

$$3\sqrt{2}k-4k=-6\sqrt{2}+18$$

Combine all terms containing k.

$$\left(3\sqrt{2}-4\right)k=-6\sqrt{2}+18$$

The equation is in standard form.

$$\left(3\sqrt{2}-4\right)k=18-6\sqrt{2}$$

Divide both sides by 3\sqrt{2}-4.

$$\frac{\left(3\sqrt{2}-4\right)k}{3\sqrt{2}-4}=\frac{18-6\sqrt{2}}{3\sqrt{2}-4}$$

Dividing by 3\sqrt{2}-4 undoes the multiplication by 3\sqrt{2}-4.

$$k=\frac{18-6\sqrt{2}}{3\sqrt{2}-4}$$

Divide -6\sqrt{2}+18 by 3\sqrt{2}-4.

$$k=15\sqrt{2}+18$$