Solve 1/3+2sqrt{2}=(3-2sqrt{2})x+12sqrt{2}

$$\frac { 1 } { 3 + 2 \sqrt { 2 } } = ( 3 - 2 \sqrt { 2 } ) x + 12 \sqrt { 2 }$$

Solve for x

$$x=-36\sqrt{2}-47\approx -97.911688245$$

Steps for Solving Linear Equation

Rationalize the denominator of \frac{1}{3+2\sqrt{2}} by multiplying numerator and denominator by 3-2\sqrt{2}.

$$\frac{3-2\sqrt{2}}{\left(3+2\sqrt{2}\right)\left(3-2\sqrt{2}\right)}=\left(3-2\sqrt{2}\right)x+12\sqrt{2}$$

Consider \left(3+2\sqrt{2}\right)\left(3-2\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}.

$$\frac{3-2\sqrt{2}}{3^{2}-\left(2\sqrt{2}\right)^{2}}=\left(3-2\sqrt{2}\right)x+12\sqrt{2}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{3-2\sqrt{2}}{9-\left(2\sqrt{2}\right)^{2}}=\left(3-2\sqrt{2}\right)x+12\sqrt{2}$$

Expand \left(2\sqrt{2}\right)^{2}.

$$\frac{3-2\sqrt{2}}{9-2^{2}\left(\sqrt{2}\right)^{2}}=\left(3-2\sqrt{2}\right)x+12\sqrt{2}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{3-2\sqrt{2}}{9-4\left(\sqrt{2}\right)^{2}}=\left(3-2\sqrt{2}\right)x+12\sqrt{2}$$

The square of \sqrt{2} is 2.

$$\frac{3-2\sqrt{2}}{9-4\times 2}=\left(3-2\sqrt{2}\right)x+12\sqrt{2}$$

Multiply 4 and 2 to get 8.

$$\frac{3-2\sqrt{2}}{9-8}=\left(3-2\sqrt{2}\right)x+12\sqrt{2}$$

Subtract 8 from 9 to get 1.

$$\frac{3-2\sqrt{2}}{1}=\left(3-2\sqrt{2}\right)x+12\sqrt{2}$$

Anything divided by one gives itself.

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

Use the distributive property to multiply 3-2\sqrt{2} by x.

$$3-2\sqrt{2}=3x-2\sqrt{2}x+12\sqrt{2}$$

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

$$3x-2\sqrt{2}x+12\sqrt{2}=3-2\sqrt{2}$$

Subtract 12\sqrt{2} from both sides.

$$3x-2\sqrt{2}x=3-2\sqrt{2}-12\sqrt{2}$$

Combine -2\sqrt{2} and -12\sqrt{2} to get -14\sqrt{2}.

$$3x-2\sqrt{2}x=3-14\sqrt{2}$$

Combine all terms containing x.

$$\left(3-2\sqrt{2}\right)x=3-14\sqrt{2}$$

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

$$\frac{\left(3-2\sqrt{2}\right)x}{3-2\sqrt{2}}=\frac{3-14\sqrt{2}}{3-2\sqrt{2}}$$

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

$$x=\frac{3-14\sqrt{2}}{3-2\sqrt{2}}$$

Divide -14\sqrt{2}+3 by 3-2\sqrt{2}.

$$x=-36\sqrt{2}-47$$