Solve sqrt{3/5}(x+1)+sqrt{5/3}(x-1)=1/15

$$\sqrt { \frac { 3 } { 5 } } ( x + 1 ) + \sqrt { \frac { 5 } { 3 } } ( x - 1 ) = \frac { 1 } { 15 }$$

Solve for x

$$x=\frac{\sqrt{15}+30}{120}\approx 0.282274861$$

Steps for Solving Linear Equation

Rewrite the square root of the division \sqrt{\frac{3}{5}} as the division of square roots \frac{\sqrt{3}}{\sqrt{5}}.

$$\frac{\sqrt{3}}{\sqrt{5}}\left(x+1\right)+\sqrt{\frac{5}{3}}\left(x-1\right)=\frac{1}{15}$$

Rationalize the denominator of \frac{\sqrt{3}}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{\sqrt{3}\sqrt{5}}{\left(\sqrt{5}\right)^{2}}\left(x+1\right)+\sqrt{\frac{5}{3}}\left(x-1\right)=\frac{1}{15}$$

The square of \sqrt{5} is 5.

$$\frac{\sqrt{3}\sqrt{5}}{5}\left(x+1\right)+\sqrt{\frac{5}{3}}\left(x-1\right)=\frac{1}{15}$$

To multiply \sqrt{3} and \sqrt{5}, multiply the numbers under the square root.

$$\frac{\sqrt{15}}{5}\left(x+1\right)+\sqrt{\frac{5}{3}}\left(x-1\right)=\frac{1}{15}$$

Express \frac{\sqrt{15}}{5}\left(x+1\right) as a single fraction.

$$\frac{\sqrt{15}\left(x+1\right)}{5}+\sqrt{\frac{5}{3}}\left(x-1\right)=\frac{1}{15}$$

Rewrite the square root of the division \sqrt{\frac{5}{3}} as the division of square roots \frac{\sqrt{5}}{\sqrt{3}}.

$$\frac{\sqrt{15}\left(x+1\right)}{5}+\frac{\sqrt{5}}{\sqrt{3}}\left(x-1\right)=\frac{1}{15}$$

Rationalize the denominator of \frac{\sqrt{5}}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{\sqrt{15}\left(x+1\right)}{5}+\frac{\sqrt{5}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}\left(x-1\right)=\frac{1}{15}$$

The square of \sqrt{3} is 3.

$$\frac{\sqrt{15}\left(x+1\right)}{5}+\frac{\sqrt{5}\sqrt{3}}{3}\left(x-1\right)=\frac{1}{15}$$

To multiply \sqrt{5} and \sqrt{3}, multiply the numbers under the square root.

$$\frac{\sqrt{15}\left(x+1\right)}{5}+\frac{\sqrt{15}}{3}\left(x-1\right)=\frac{1}{15}$$

Express \frac{\sqrt{15}}{3}\left(x-1\right) as a single fraction.

$$\frac{\sqrt{15}\left(x+1\right)}{5}+\frac{\sqrt{15}\left(x-1\right)}{3}=\frac{1}{15}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 5 and 3 is 15. Multiply \frac{\sqrt{15}\left(x+1\right)}{5} times \frac{3}{3}. Multiply \frac{\sqrt{15}\left(x-1\right)}{3} times \frac{5}{5}.

$$\frac{3\sqrt{15}\left(x+1\right)}{15}+\frac{5\sqrt{15}\left(x-1\right)}{15}=\frac{1}{15}$$

Since \frac{3\sqrt{15}\left(x+1\right)}{15} and \frac{5\sqrt{15}\left(x-1\right)}{15} have the same denominator, add them by adding their numerators.

$$\frac{3\sqrt{15}\left(x+1\right)+5\sqrt{15}\left(x-1\right)}{15}=\frac{1}{15}$$

Do the multiplications in 3\sqrt{15}\left(x+1\right)+5\sqrt{15}\left(x-1\right).

$$\frac{3\sqrt{15}x+3\sqrt{15}+5\sqrt{15}x-5\sqrt{15}}{15}=\frac{1}{15}$$

Combine like terms in 3\sqrt{15}x+3\sqrt{15}+5\sqrt{15}x-5\sqrt{15}.

$$\frac{8\sqrt{15}x-2\sqrt{15}}{15}=\frac{1}{15}$$

Multiply both sides by 15.

$$8\sqrt{15}x-2\sqrt{15}=\frac{1}{15}\times 15$$

Cancel out 15 and 15.

$$8\sqrt{15}x-2\sqrt{15}=1$$

Add 2\sqrt{15} to both sides.

$$8\sqrt{15}x=1+2\sqrt{15}$$

The equation is in standard form.

$$8\sqrt{15}x=2\sqrt{15}+1$$

Divide both sides by 8\sqrt{15}.

$$\frac{8\sqrt{15}x}{8\sqrt{15}}=\frac{2\sqrt{15}+1}{8\sqrt{15}}$$

Dividing by 8\sqrt{15} undoes the multiplication by 8\sqrt{15}.

$$x=\frac{2\sqrt{15}+1}{8\sqrt{15}}$$

Divide 1+2\sqrt{15} by 8\sqrt{15}.

$$x=\frac{\sqrt{15}}{120}+\frac{1}{4}$$