Solve frac{5x-2sqrt{2}}{sqrt{2}}-x-3/2-11x-1/3sqrt{2}=0

$$\frac { 5 x - 2 \sqrt { 2 } } { \sqrt { 2 } } - \frac { x - 3 } { 2 } - \frac { 11 x - 1 } { 3 \sqrt { 2 } } = 0$$

Solve for x

$$x=\frac{9\sqrt{2}+1}{23}\approx 0.596866177$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 2.

$$2^{\frac{1}{2}}\left(5x-2\sqrt{2}\right)-\left(x-3\right)-\frac{1}{3}\times 2^{\frac{1}{2}}\left(11x-1\right)=0$$

Use the distributive property to multiply 2^{\frac{1}{2}} by 5x-2\sqrt{2}.

$$2^{\frac{1}{2}}\times 5x+2^{\frac{1}{2}}\left(-2\right)\sqrt{2}-\left(x-3\right)-\frac{1}{3}\times 2^{\frac{1}{2}}\left(11x-1\right)=0$$

To find the opposite of x-3, find the opposite of each term.

$$2^{\frac{1}{2}}\times 5x+2^{\frac{1}{2}}\left(-2\right)\sqrt{2}-x-\left(-3\right)-\frac{1}{3}\times 2^{\frac{1}{2}}\left(11x-1\right)=0$$

The opposite of -3 is 3.

$$2^{\frac{1}{2}}\times 5x+2^{\frac{1}{2}}\left(-2\right)\sqrt{2}-x+3-\frac{1}{3}\times 2^{\frac{1}{2}}\left(11x-1\right)=0$$

Use the distributive property to multiply \frac{1}{3}\times 2^{\frac{1}{2}} by 11x-1.

$$2^{\frac{1}{2}}\times 5x+2^{\frac{1}{2}}\left(-2\right)\sqrt{2}-x+3-\left(\frac{1}{3}\times 2^{\frac{1}{2}}\times 11x+\frac{1}{3}\times 2^{\frac{1}{2}}\left(-1\right)\right)=0$$

Multiply \frac{1}{3} and 11 to get \frac{11}{3}.

$$2^{\frac{1}{2}}\times 5x+2^{\frac{1}{2}}\left(-2\right)\sqrt{2}-x+3-\left(\frac{11}{3}\times 2^{\frac{1}{2}}x+\frac{1}{3}\times 2^{\frac{1}{2}}\left(-1\right)\right)=0$$

Multiply \frac{1}{3} and -1 to get -\frac{1}{3}.

$$2^{\frac{1}{2}}\times 5x+2^{\frac{1}{2}}\left(-2\right)\sqrt{2}-x+3-\left(\frac{11}{3}\times 2^{\frac{1}{2}}x-\frac{1}{3}\times 2^{\frac{1}{2}}\right)=0$$

To find the opposite of \frac{11}{3}\times 2^{\frac{1}{2}}x-\frac{1}{3}\times 2^{\frac{1}{2}}, find the opposite of each term.

$$2^{\frac{1}{2}}\times 5x+2^{\frac{1}{2}}\left(-2\right)\sqrt{2}-x+3-\frac{11}{3}\times 2^{\frac{1}{2}}x-\left(-\frac{1}{3}\times 2^{\frac{1}{2}}\right)=0$$

The opposite of -\frac{1}{3}\times 2^{\frac{1}{2}} is \frac{1}{3}\times 2^{\frac{1}{2}}.

$$2^{\frac{1}{2}}\times 5x+2^{\frac{1}{2}}\left(-2\right)\sqrt{2}-x+3-\frac{11}{3}\times 2^{\frac{1}{2}}x+\frac{1}{3}\times 2^{\frac{1}{2}}=0$$

Combine 2^{\frac{1}{2}}\times 5x and -\frac{11}{3}\times 2^{\frac{1}{2}}x to get \frac{4}{3}\times 2^{\frac{1}{2}}x.

$$\frac{4}{3}\times 2^{\frac{1}{2}}x+2^{\frac{1}{2}}\left(-2\right)\sqrt{2}-x+3+\frac{1}{3}\times 2^{\frac{1}{2}}=0$$

Subtract 2^{\frac{1}{2}}\left(-2\right)\sqrt{2} from both sides. Anything subtracted from zero gives its negation.

$$\frac{4}{3}\times 2^{\frac{1}{2}}x-x+3+\frac{1}{3}\times 2^{\frac{1}{2}}=-2^{\frac{1}{2}}\left(-2\right)\sqrt{2}$$

Subtract 3 from both sides.

$$\frac{4}{3}\times 2^{\frac{1}{2}}x-x+\frac{1}{3}\times 2^{\frac{1}{2}}=-2^{\frac{1}{2}}\left(-2\right)\sqrt{2}-3$$

Subtract \frac{1}{3}\times 2^{\frac{1}{2}} from both sides.

$$\frac{4}{3}\times 2^{\frac{1}{2}}x-x=-2^{\frac{1}{2}}\left(-2\right)\sqrt{2}-3-\frac{1}{3}\times 2^{\frac{1}{2}}$$

Reorder the terms.

$$\frac{4}{3}\sqrt{2}x-x=-\left(-2\sqrt{2}\sqrt{2}\right)-3-\frac{1}{3}\sqrt{2}$$

Multiply \sqrt{2} and \sqrt{2} to get 2.

$$\frac{4}{3}\sqrt{2}x-x=-\left(-2\right)\times 2-3-\frac{1}{3}\sqrt{2}$$

Multiply -1 and -2 to get 2.

$$\frac{4}{3}\sqrt{2}x-x=2\times 2-3-\frac{1}{3}\sqrt{2}$$

Multiply 2 and 2 to get 4.

$$\frac{4}{3}\sqrt{2}x-x=4-3-\frac{1}{3}\sqrt{2}$$

Subtract 3 from 4 to get 1.

$$\frac{4}{3}\sqrt{2}x-x=1-\frac{1}{3}\sqrt{2}$$

Combine all terms containing x.

$$\left(\frac{4}{3}\sqrt{2}-1\right)x=1-\frac{1}{3}\sqrt{2}$$

The equation is in standard form.

$$\left(\frac{4\sqrt{2}}{3}-1\right)x=-\frac{\sqrt{2}}{3}+1$$

Divide both sides by \frac{4}{3}\sqrt{2}-1.

$$\frac{\left(\frac{4\sqrt{2}}{3}-1\right)x}{\frac{4\sqrt{2}}{3}-1}=\frac{-\frac{\sqrt{2}}{3}+1}{\frac{4\sqrt{2}}{3}-1}$$

Dividing by \frac{4}{3}\sqrt{2}-1 undoes the multiplication by \frac{4}{3}\sqrt{2}-1.

$$x=\frac{-\frac{\sqrt{2}}{3}+1}{\frac{4\sqrt{2}}{3}-1}$$

Divide 1-\frac{\sqrt{2}}{3} by \frac{4}{3}\sqrt{2}-1.

$$x=\frac{9\sqrt{2}+1}{23}$$