Solve 1/2(1-1/2+1/sqrt{8}y)+2y*1/18=1

$$\frac { 1 } { 2 } ( 1 - \frac { 1 } { 2 } + \frac { 1 } { \sqrt { 8 } } y ) + 2 y \times \frac { 1 } { 18 } = 1$$

Solve for y

$$y = \frac{243 \sqrt{2} - 216}{49} \approx 2.605181544$$

Steps for Solving Linear Equation

Convert 1 to fraction \frac{2}{2}.

$$\frac{1}{2}\left(\frac{2}{2}-\frac{1}{2}+\frac{1}{\sqrt{8}}y\right)+2y\times \frac{1}{18}=1$$

Since \frac{2}{2} and \frac{1}{2} have the same denominator, subtract them by subtracting their numerators.

$$\frac{1}{2}\left(\frac{2-1}{2}+\frac{1}{\sqrt{8}}y\right)+2y\times \frac{1}{18}=1$$

Subtract 1 from 2 to get 1.

$$\frac{1}{2}\left(\frac{1}{2}+\frac{1}{\sqrt{8}}y\right)+2y\times \frac{1}{18}=1$$

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}.

$$\frac{1}{2}\left(\frac{1}{2}+\frac{1}{2\sqrt{2}}y\right)+2y\times \frac{1}{18}=1$$

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

$$\frac{1}{2}\left(\frac{1}{2}+\frac{\sqrt{2}}{2\left(\sqrt{2}\right)^{2}}y\right)+2y\times \frac{1}{18}=1$$

The square of \sqrt{2} is 2.

$$\frac{1}{2}\left(\frac{1}{2}+\frac{\sqrt{2}}{2\times 2}y\right)+2y\times \frac{1}{18}=1$$

Multiply 2 and 2 to get 4.

$$\frac{1}{2}\left(\frac{1}{2}+\frac{\sqrt{2}}{4}y\right)+2y\times \frac{1}{18}=1$$

Express \frac{\sqrt{2}}{4}y as a single fraction.

$$\frac{1}{2}\left(\frac{1}{2}+\frac{\sqrt{2}y}{4}\right)+2y\times \frac{1}{18}=1$$

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

$$\frac{1}{2}\left(\frac{2}{4}+\frac{\sqrt{2}y}{4}\right)+2y\times \frac{1}{18}=1$$

Since \frac{2}{4} and \frac{\sqrt{2}y}{4} have the same denominator, add them by adding their numerators.

$$\frac{1}{2}\times \frac{2+\sqrt{2}y}{4}+2y\times \frac{1}{18}=1$$

Multiply \frac{1}{2} times \frac{2+\sqrt{2}y}{4} by multiplying numerator times numerator and denominator times denominator.

$$\frac{2+\sqrt{2}y}{2\times 4}+2y\times \frac{1}{18}=1$$

Multiply 2 and \frac{1}{18} to get \frac{2}{18}.

$$\frac{2+\sqrt{2}y}{2\times 4}+\frac{2}{18}y=1$$

Reduce the fraction \frac{2}{18} to lowest terms by extracting and canceling out 2.

$$\frac{2+\sqrt{2}y}{2\times 4}+\frac{1}{9}y=1$$

Multiply 2 and 4 to get 8.

$$\frac{2+\sqrt{2}y}{8}+\frac{1}{9}y=1$$

Multiply both sides of the equation by 72, the least common multiple of 8,9.

$$9\left(2+\sqrt{2}y\right)+8y=72$$

Use the distributive property to multiply 9 by 2+\sqrt{2}y.

$$18+9\sqrt{2}y+8y=72$$

Subtract 18 from both sides.

$$9\sqrt{2}y+8y=72-18$$

Subtract 18 from 72 to get 54.

$$9\sqrt{2}y+8y=54$$

Combine all terms containing y.

$$\left(9\sqrt{2}+8\right)y=54$$

Divide both sides by 9\sqrt{2}+8.

$$\frac{\left(9\sqrt{2}+8\right)y}{9\sqrt{2}+8}=\frac{54}{9\sqrt{2}+8}$$

Dividing by 9\sqrt{2}+8 undoes the multiplication by 9\sqrt{2}+8.

$$y=\frac{54}{9\sqrt{2}+8}$$

Divide 54 by 9\sqrt{2}+8.

$$y=\frac{243\sqrt{2}-216}{49}$$