Solve 2y/3=3/5+2/y

$$\frac { 2 y } { 3 } = \frac { 3 } { 5 } + \frac { 2 } { y }$$

Solve for y

$$y = \frac{\sqrt{1281} + 9}{20} \approx 2.239553017$$

$$y=\frac{9-\sqrt{1281}}{20}\approx -1.339553017$$

Steps Using the Quadratic Formula

Variable y cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 15y, the least common multiple of 3,5,y.

$$5y\times 2y=15y\times \frac{3}{5}+15\times 2$$

Multiply 5 and 2 to get 10.

$$10yy=15y\times \frac{3}{5}+15\times 2$$

Multiply y and y to get y^{2}.

$$10y^{2}=15y\times \frac{3}{5}+15\times 2$$

Express 15\times \frac{3}{5} as a single fraction.

$$10y^{2}=\frac{15\times 3}{5}y+15\times 2$$

Multiply 15 and 3 to get 45.

$$10y^{2}=\frac{45}{5}y+15\times 2$$

Divide 45 by 5 to get 9.

$$10y^{2}=9y+15\times 2$$

Multiply 15 and 2 to get 30.

$$10y^{2}=9y+30$$

Subtract 9y from both sides.

$$10y^{2}-9y=30$$

Subtract 30 from both sides.

$$10y^{2}-9y-30=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 10 for a, -9 for b, and -30 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\times 10\left(-30\right)}}{2\times 10}$$

Square -9.

$$y=\frac{-\left(-9\right)±\sqrt{81-4\times 10\left(-30\right)}}{2\times 10}$$

Multiply -4 times 10.

$$y=\frac{-\left(-9\right)±\sqrt{81-40\left(-30\right)}}{2\times 10}$$

Multiply -40 times -30.

$$y=\frac{-\left(-9\right)±\sqrt{81+1200}}{2\times 10}$$

Add 81 to 1200.

$$y=\frac{-\left(-9\right)±\sqrt{1281}}{2\times 10}$$

The opposite of -9 is 9.

$$y=\frac{9±\sqrt{1281}}{2\times 10}$$

Multiply 2 times 10.

$$y=\frac{9±\sqrt{1281}}{20}$$

Now solve the equation y=\frac{9±\sqrt{1281}}{20} when ± is plus. Add 9 to \sqrt{1281}.

$$y=\frac{\sqrt{1281}+9}{20}$$

Now solve the equation y=\frac{9±\sqrt{1281}}{20} when ± is minus. Subtract \sqrt{1281} from 9.

$$y=\frac{9-\sqrt{1281}}{20}$$

The equation is now solved.

$$y=\frac{\sqrt{1281}+9}{20} y=\frac{9-\sqrt{1281}}{20}$$

Steps for Completing the Square

Variable y cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 15y, the least common multiple of 3,5,y.

$$5y\times 2y=15y\times \frac{3}{5}+15\times 2$$

Multiply 5 and 2 to get 10.

$$10yy=15y\times \frac{3}{5}+15\times 2$$

Multiply y and y to get y^{2}.

$$10y^{2}=15y\times \frac{3}{5}+15\times 2$$

Express 15\times \frac{3}{5} as a single fraction.

$$10y^{2}=\frac{15\times 3}{5}y+15\times 2$$

Multiply 15 and 3 to get 45.

$$10y^{2}=\frac{45}{5}y+15\times 2$$

Divide 45 by 5 to get 9.

$$10y^{2}=9y+15\times 2$$

Multiply 15 and 2 to get 30.

$$10y^{2}=9y+30$$

Subtract 9y from both sides.

$$10y^{2}-9y=30$$

Divide both sides by 10.

$$\frac{10y^{2}-9y}{10}=\frac{30}{10}$$

Dividing by 10 undoes the multiplication by 10.

$$y^{2}-\frac{9}{10}y=\frac{30}{10}$$

Divide 30 by 10.

$$y^{2}-\frac{9}{10}y=3$$

Divide -\frac{9}{10}, the coefficient of the x term, by 2 to get -\frac{9}{20}. Then add the square of -\frac{9}{20} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}-\frac{9}{10}y+\left(-\frac{9}{20}\right)^{2}=3+\left(-\frac{9}{20}\right)^{2}$$

Square -\frac{9}{20} by squaring both the numerator and the denominator of the fraction.

$$y^{2}-\frac{9}{10}y+\frac{81}{400}=3+\frac{81}{400}$$

Add 3 to \frac{81}{400}.

$$y^{2}-\frac{9}{10}y+\frac{81}{400}=\frac{1281}{400}$$

Factor y^{2}-\frac{9}{10}y+\frac{81}{400}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y-\frac{9}{20}\right)^{2}=\frac{1281}{400}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-\frac{9}{20}\right)^{2}}=\sqrt{\frac{1281}{400}}$$

Simplify.

$$y-\frac{9}{20}=\frac{\sqrt{1281}}{20} y-\frac{9}{20}=-\frac{\sqrt{1281}}{20}$$

Add \frac{9}{20} to both sides of the equation.

$$y=\frac{\sqrt{1281}+9}{20} y=\frac{9-\sqrt{1281}}{20}$$