Solve y/y+5-5y+9/y^2+10y+25-2y/3y+15=

$$\frac { y } { y + 5 } - \frac { 5 y + 9 } { y ^ { 2 } + 10 y + 25 } - \frac { 2 y } { 3 y + 15 } =$$

Evaluate

$$\frac{y^{2}-10y-27}{3\left(y+5\right)^{2}}$$

Short Solution Steps

Factor y^{2}+10y+25.

$$\frac{y}{y+5}-\frac{5y+9}{\left(y+5\right)^{2}}-\frac{2y}{3y+15}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of y+5 and \left(y+5\right)^{2} is \left(y+5\right)^{2}. Multiply \frac{y}{y+5} times \frac{y+5}{y+5}.

$$\frac{y\left(y+5\right)}{\left(y+5\right)^{2}}-\frac{5y+9}{\left(y+5\right)^{2}}-\frac{2y}{3y+15}$$

Since \frac{y\left(y+5\right)}{\left(y+5\right)^{2}} and \frac{5y+9}{\left(y+5\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{y\left(y+5\right)-\left(5y+9\right)}{\left(y+5\right)^{2}}-\frac{2y}{3y+15}$$

Do the multiplications in y\left(y+5\right)-\left(5y+9\right).

$$\frac{y^{2}+5y-5y-9}{\left(y+5\right)^{2}}-\frac{2y}{3y+15}$$

Combine like terms in y^{2}+5y-5y-9.

$$\frac{y^{2}-9}{\left(y+5\right)^{2}}-\frac{2y}{3y+15}$$

Factor 3y+15.

$$\frac{y^{2}-9}{\left(y+5\right)^{2}}-\frac{2y}{3\left(y+5\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(y+5\right)^{2} and 3\left(y+5\right) is 3\left(y+5\right)^{2}. Multiply \frac{y^{2}-9}{\left(y+5\right)^{2}} times \frac{3}{3}. Multiply \frac{2y}{3\left(y+5\right)} times \frac{y+5}{y+5}.

$$\frac{3\left(y^{2}-9\right)}{3\left(y+5\right)^{2}}-\frac{2y\left(y+5\right)}{3\left(y+5\right)^{2}}$$

Since \frac{3\left(y^{2}-9\right)}{3\left(y+5\right)^{2}} and \frac{2y\left(y+5\right)}{3\left(y+5\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{3\left(y^{2}-9\right)-2y\left(y+5\right)}{3\left(y+5\right)^{2}}$$

Do the multiplications in 3\left(y^{2}-9\right)-2y\left(y+5\right).

$$\frac{3y^{2}-27-2y^{2}-10y}{3\left(y+5\right)^{2}}$$

Combine like terms in 3y^{2}-27-2y^{2}-10y.

$$\frac{y^{2}-27-10y}{3\left(y+5\right)^{2}}$$

Expand 3\left(y+5\right)^{2}.

$$\frac{y^{2}-27-10y}{3y^{2}+30y+75}$$

Expand

$$\frac{y^{2}-10y-27}{3\left(y+5\right)^{2}}$$

Short Solution Steps

Factor y^{2}+10y+25.

$$\frac{y}{y+5}-\frac{5y+9}{\left(y+5\right)^{2}}-\frac{2y}{3y+15}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of y+5 and \left(y+5\right)^{2} is \left(y+5\right)^{2}. Multiply \frac{y}{y+5} times \frac{y+5}{y+5}.

$$\frac{y\left(y+5\right)}{\left(y+5\right)^{2}}-\frac{5y+9}{\left(y+5\right)^{2}}-\frac{2y}{3y+15}$$

Since \frac{y\left(y+5\right)}{\left(y+5\right)^{2}} and \frac{5y+9}{\left(y+5\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{y\left(y+5\right)-\left(5y+9\right)}{\left(y+5\right)^{2}}-\frac{2y}{3y+15}$$

Do the multiplications in y\left(y+5\right)-\left(5y+9\right).

$$\frac{y^{2}+5y-5y-9}{\left(y+5\right)^{2}}-\frac{2y}{3y+15}$$

Combine like terms in y^{2}+5y-5y-9.

$$\frac{y^{2}-9}{\left(y+5\right)^{2}}-\frac{2y}{3y+15}$$

Factor 3y+15.

$$\frac{y^{2}-9}{\left(y+5\right)^{2}}-\frac{2y}{3\left(y+5\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(y+5\right)^{2} and 3\left(y+5\right) is 3\left(y+5\right)^{2}. Multiply \frac{y^{2}-9}{\left(y+5\right)^{2}} times \frac{3}{3}. Multiply \frac{2y}{3\left(y+5\right)} times \frac{y+5}{y+5}.

$$\frac{3\left(y^{2}-9\right)}{3\left(y+5\right)^{2}}-\frac{2y\left(y+5\right)}{3\left(y+5\right)^{2}}$$

Since \frac{3\left(y^{2}-9\right)}{3\left(y+5\right)^{2}} and \frac{2y\left(y+5\right)}{3\left(y+5\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{3\left(y^{2}-9\right)-2y\left(y+5\right)}{3\left(y+5\right)^{2}}$$

Do the multiplications in 3\left(y^{2}-9\right)-2y\left(y+5\right).

$$\frac{3y^{2}-27-2y^{2}-10y}{3\left(y+5\right)^{2}}$$

Combine like terms in 3y^{2}-27-2y^{2}-10y.

$$\frac{y^{2}-27-10y}{3\left(y+5\right)^{2}}$$

Expand 3\left(y+5\right)^{2}.

$$\frac{y^{2}-27-10y}{3y^{2}+30y+75}$$