Solve y=5(10-y)/3(y+1)

$$y = \frac { 5 ( 10 - y ) } { 3 ( y + 1 ) }$$

Solve for y

$$y = \frac{\sqrt{166} - 4}{3} \approx 2.961366242$$

$$y=\frac{-\sqrt{166}-4}{3}\approx -5.628032909$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 5 by 10-y.

$$y=\frac{50-5y}{3\left(y+1\right)}$$

Use the distributive property to multiply 3 by y+1.

$$y=\frac{50-5y}{3y+3}$$

Subtract \frac{50-5y}{3y+3} from both sides.

$$y-\frac{50-5y}{3y+3}=0$$

Factor 3y+3.

$$y-\frac{50-5y}{3\left(y+1\right)}=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply y times \frac{3\left(y+1\right)}{3\left(y+1\right)}.

$$\frac{y\times 3\left(y+1\right)}{3\left(y+1\right)}-\frac{50-5y}{3\left(y+1\right)}=0$$

Since \frac{y\times 3\left(y+1\right)}{3\left(y+1\right)} and \frac{50-5y}{3\left(y+1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{y\times 3\left(y+1\right)-\left(50-5y\right)}{3\left(y+1\right)}=0$$

Do the multiplications in y\times 3\left(y+1\right)-\left(50-5y\right).

$$\frac{3y^{2}+3y-50+5y}{3\left(y+1\right)}=0$$

Combine like terms in 3y^{2}+3y-50+5y.

$$\frac{3y^{2}+8y-50}{3\left(y+1\right)}=0$$

Factor the expressions that are not already factored in \frac{3y^{2}+8y-50}{3\left(y+1\right)}.

$$\frac{3\left(y-\left(-\frac{1}{3}\sqrt{166}-\frac{4}{3}\right)\right)\left(y-\left(\frac{1}{3}\sqrt{166}-\frac{4}{3}\right)\right)}{3\left(y+1\right)}=0$$

Cancel out 3 in both numerator and denominator.

$$\frac{\left(y-\left(-\frac{1}{3}\sqrt{166}-\frac{4}{3}\right)\right)\left(y-\left(\frac{1}{3}\sqrt{166}-\frac{4}{3}\right)\right)}{y+1}=0$$

Variable y cannot be equal to -1 since division by zero is not defined. Multiply both sides of the equation by y+1.

$$\left(y-\left(-\frac{1}{3}\sqrt{166}-\frac{4}{3}\right)\right)\left(y-\left(\frac{1}{3}\sqrt{166}-\frac{4}{3}\right)\right)=0$$

To find the opposite of -\frac{1}{3}\sqrt{166}-\frac{4}{3}, find the opposite of each term.

$$\left(y+\frac{1}{3}\sqrt{166}+\frac{4}{3}\right)\left(y-\left(\frac{1}{3}\sqrt{166}-\frac{4}{3}\right)\right)=0$$

To find the opposite of \frac{1}{3}\sqrt{166}-\frac{4}{3}, find the opposite of each term.

$$\left(y+\frac{1}{3}\sqrt{166}+\frac{4}{3}\right)\left(y-\frac{1}{3}\sqrt{166}+\frac{4}{3}\right)=0$$

Use the distributive property to multiply y+\frac{1}{3}\sqrt{166}+\frac{4}{3} by y-\frac{1}{3}\sqrt{166}+\frac{4}{3} and combine like terms.

$$y^{2}+\frac{8}{3}y-\frac{1}{9}\left(\sqrt{166}\right)^{2}+\frac{16}{9}=0$$

The square of \sqrt{166} is 166.

$$y^{2}+\frac{8}{3}y-\frac{1}{9}\times 166+\frac{16}{9}=0$$

Multiply -\frac{1}{9} and 166 to get -\frac{166}{9}.

$$y^{2}+\frac{8}{3}y-\frac{166}{9}+\frac{16}{9}=0$$

Add -\frac{166}{9} and \frac{16}{9} to get -\frac{50}{3}.

$$y^{2}+\frac{8}{3}y-\frac{50}{3}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, \frac{8}{3} for b, and -\frac{50}{3} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\frac{8}{3}±\sqrt{\left(\frac{8}{3}\right)^{2}-4\left(-\frac{50}{3}\right)}}{2}$$

Square \frac{8}{3} by squaring both the numerator and the denominator of the fraction.

$$y=\frac{-\frac{8}{3}±\sqrt{\frac{64}{9}-4\left(-\frac{50}{3}\right)}}{2}$$

Multiply -4 times -\frac{50}{3}.

$$y=\frac{-\frac{8}{3}±\sqrt{\frac{64}{9}+\frac{200}{3}}}{2}$$

Add \frac{64}{9} to \frac{200}{3} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y=\frac{-\frac{8}{3}±\sqrt{\frac{664}{9}}}{2}$$

Take the square root of \frac{664}{9}.

$$y=\frac{-\frac{8}{3}±\frac{2\sqrt{166}}{3}}{2}$$

Now solve the equation y=\frac{-\frac{8}{3}±\frac{2\sqrt{166}}{3}}{2} when ± is plus. Add -\frac{8}{3} to \frac{2\sqrt{166}}{3}.

$$y=\frac{2\sqrt{166}-8}{2\times 3}$$

Divide \frac{-8+2\sqrt{166}}{3} by 2.

$$y=\frac{\sqrt{166}-4}{3}$$

Now solve the equation y=\frac{-\frac{8}{3}±\frac{2\sqrt{166}}{3}}{2} when ± is minus. Subtract \frac{2\sqrt{166}}{3} from -\frac{8}{3}.

$$y=\frac{-2\sqrt{166}-8}{2\times 3}$$

Divide \frac{-8-2\sqrt{166}}{3} by 2.

$$y=\frac{-\sqrt{166}-4}{3}$$

The equation is now solved.

$$y=\frac{\sqrt{166}-4}{3} y=\frac{-\sqrt{166}-4}{3}$$

Steps for Completing the Square

Use the distributive property to multiply 5 by 10-y.

$$y=\frac{50-5y}{3\left(y+1\right)}$$

Use the distributive property to multiply 3 by y+1.

$$y=\frac{50-5y}{3y+3}$$

Subtract \frac{50-5y}{3y+3} from both sides.

$$y-\frac{50-5y}{3y+3}=0$$

Factor 3y+3.

$$y-\frac{50-5y}{3\left(y+1\right)}=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply y times \frac{3\left(y+1\right)}{3\left(y+1\right)}.

$$\frac{y\times 3\left(y+1\right)}{3\left(y+1\right)}-\frac{50-5y}{3\left(y+1\right)}=0$$

Since \frac{y\times 3\left(y+1\right)}{3\left(y+1\right)} and \frac{50-5y}{3\left(y+1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{y\times 3\left(y+1\right)-\left(50-5y\right)}{3\left(y+1\right)}=0$$

Do the multiplications in y\times 3\left(y+1\right)-\left(50-5y\right).

$$\frac{3y^{2}+3y-50+5y}{3\left(y+1\right)}=0$$

Combine like terms in 3y^{2}+3y-50+5y.

$$\frac{3y^{2}+8y-50}{3\left(y+1\right)}=0$$

Factor the expressions that are not already factored in \frac{3y^{2}+8y-50}{3\left(y+1\right)}.

$$\frac{3\left(y-\left(-\frac{1}{3}\sqrt{166}-\frac{4}{3}\right)\right)\left(y-\left(\frac{1}{3}\sqrt{166}-\frac{4}{3}\right)\right)}{3\left(y+1\right)}=0$$

Cancel out 3 in both numerator and denominator.

$$\frac{\left(y-\left(-\frac{1}{3}\sqrt{166}-\frac{4}{3}\right)\right)\left(y-\left(\frac{1}{3}\sqrt{166}-\frac{4}{3}\right)\right)}{y+1}=0$$

Variable y cannot be equal to -1 since division by zero is not defined. Multiply both sides of the equation by y+1.

$$\left(y-\left(-\frac{1}{3}\sqrt{166}-\frac{4}{3}\right)\right)\left(y-\left(\frac{1}{3}\sqrt{166}-\frac{4}{3}\right)\right)=0$$

To find the opposite of -\frac{1}{3}\sqrt{166}-\frac{4}{3}, find the opposite of each term.

$$\left(y+\frac{1}{3}\sqrt{166}+\frac{4}{3}\right)\left(y-\left(\frac{1}{3}\sqrt{166}-\frac{4}{3}\right)\right)=0$$

To find the opposite of \frac{1}{3}\sqrt{166}-\frac{4}{3}, find the opposite of each term.

$$\left(y+\frac{1}{3}\sqrt{166}+\frac{4}{3}\right)\left(y-\frac{1}{3}\sqrt{166}+\frac{4}{3}\right)=0$$

Use the distributive property to multiply y+\frac{1}{3}\sqrt{166}+\frac{4}{3} by y-\frac{1}{3}\sqrt{166}+\frac{4}{3} and combine like terms.

$$y^{2}+\frac{8}{3}y-\frac{1}{9}\left(\sqrt{166}\right)^{2}+\frac{16}{9}=0$$

The square of \sqrt{166} is 166.

$$y^{2}+\frac{8}{3}y-\frac{1}{9}\times 166+\frac{16}{9}=0$$

Multiply -\frac{1}{9} and 166 to get -\frac{166}{9}.

$$y^{2}+\frac{8}{3}y-\frac{166}{9}+\frac{16}{9}=0$$

Add -\frac{166}{9} and \frac{16}{9} to get -\frac{50}{3}.

$$y^{2}+\frac{8}{3}y-\frac{50}{3}=0$$

Add \frac{50}{3} to both sides. Anything plus zero gives itself.

$$y^{2}+\frac{8}{3}y=\frac{50}{3}$$

Divide \frac{8}{3}, the coefficient of the x term, by 2 to get \frac{4}{3}. Then add the square of \frac{4}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}+\frac{8}{3}y+\left(\frac{4}{3}\right)^{2}=\frac{50}{3}+\left(\frac{4}{3}\right)^{2}$$

Square \frac{4}{3} by squaring both the numerator and the denominator of the fraction.

$$y^{2}+\frac{8}{3}y+\frac{16}{9}=\frac{50}{3}+\frac{16}{9}$$

Add \frac{50}{3} to \frac{16}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y^{2}+\frac{8}{3}y+\frac{16}{9}=\frac{166}{9}$$

Factor y^{2}+\frac{8}{3}y+\frac{16}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y+\frac{4}{3}\right)^{2}=\frac{166}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y+\frac{4}{3}\right)^{2}}=\sqrt{\frac{166}{9}}$$

Simplify.

$$y+\frac{4}{3}=\frac{\sqrt{166}}{3} y+\frac{4}{3}=-\frac{\sqrt{166}}{3}$$

Subtract \frac{4}{3} from both sides of the equation.

$$y=\frac{\sqrt{166}-4}{3} y=\frac{-\sqrt{166}-4}{3}$$