Solve 1-x/sqrt{2}-x/sqrt{8}=x-sqrt{2}

$$\frac { 1 - x } { \sqrt { 2 } } - \frac { x } { \sqrt { 8 } } = x - \sqrt { 2 }$$

Solve for x

$$x=18-12\sqrt{2}\approx 1.029437252$$

Steps for Solving Linear Equation

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

$$\frac{\left(1-x\right)\sqrt{2}}{\left(\sqrt{2}\right)^{2}}-\frac{x}{\sqrt{8}}=x-\sqrt{2}$$

The square of \sqrt{2} is 2.

$$\frac{\left(1-x\right)\sqrt{2}}{2}-\frac{x}{\sqrt{8}}=x-\sqrt{2}$$

Factor 8=2^{2}\times 2. Rewrite the square root of the product \sqrt{2^{2}\times 2} as the product of square roots \sqrt{2^{2}}\sqrt{2}. Take the square root of 2^{2}.

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

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

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

The square of \sqrt{2} is 2.

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

Multiply 2 and 2 to get 4.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2 and 4 is 4. Multiply \frac{\left(1-x\right)\sqrt{2}}{2} times \frac{2}{2}.

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

Since \frac{2\left(1-x\right)\sqrt{2}}{4} and \frac{x\sqrt{2}}{4} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 2\left(1-x\right)\sqrt{2}-x\sqrt{2}.

$$\frac{2\sqrt{2}-2x\sqrt{2}-x\sqrt{2}}{4}=x-\sqrt{2}$$

Combine like terms in 2\sqrt{2}-2x\sqrt{2}-x\sqrt{2}.

$$\frac{2\sqrt{2}-3x\sqrt{2}}{4}=x-\sqrt{2}$$

Subtract x from both sides.

$$\frac{2\sqrt{2}-3x\sqrt{2}}{4}-x=-\sqrt{2}$$

Multiply both sides of the equation by 4.

$$2\sqrt{2}-3x\sqrt{2}-4x=-4\sqrt{2}$$

Reorder the terms.

$$-3\sqrt{2}x-4x+2\sqrt{2}=-4\sqrt{2}$$

Subtract 2\sqrt{2} from both sides.

$$-3\sqrt{2}x-4x=-4\sqrt{2}-2\sqrt{2}$$

Combine -4\sqrt{2} and -2\sqrt{2} to get -6\sqrt{2}.

$$-3\sqrt{2}x-4x=-6\sqrt{2}$$

Combine all terms containing x.

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

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

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

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

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

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

$$x=18-12\sqrt{2}$$