Solve 2x/5x+bx+3y/sy+by=

$$\frac { 2 x } { 5 x + b x } + \frac { 3 y } { s y + b y } =$$

Evaluate

$$\frac{2s+5b+15}{\left(b+5\right)\left(s+b\right)}$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{2x}{5x+bx}.

$$\frac{2x}{x\left(b+5\right)}+\frac{3y}{sy+by}$$

Cancel out x in both numerator and denominator.

$$\frac{2}{b+5}+\frac{3y}{sy+by}$$

Factor the expressions that are not already factored in \frac{3y}{sy+by}.

$$\frac{2}{b+5}+\frac{3y}{y\left(b+s\right)}$$

Cancel out y in both numerator and denominator.

$$\frac{2}{b+5}+\frac{3}{s+b}$$

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

$$\frac{2\left(s+b\right)}{\left(b+5\right)\left(s+b\right)}+\frac{3\left(b+5\right)}{\left(b+5\right)\left(s+b\right)}$$

Since \frac{2\left(s+b\right)}{\left(b+5\right)\left(s+b\right)} and \frac{3\left(b+5\right)}{\left(b+5\right)\left(s+b\right)} have the same denominator, add them by adding their numerators.

$$\frac{2\left(s+b\right)+3\left(b+5\right)}{\left(b+5\right)\left(s+b\right)}$$

Do the multiplications in 2\left(s+b\right)+3\left(b+5\right).

$$\frac{2s+2b+3b+15}{\left(b+5\right)\left(s+b\right)}$$

Combine like terms in 2s+2b+3b+15.

$$\frac{2s+5b+15}{\left(b+5\right)\left(s+b\right)}$$

Expand \left(b+5\right)\left(s+b\right).

$$\frac{2s+5b+15}{bs+5s+b^{2}+5b}$$

Differentiate w.r.t. s

$$-\frac{3}{\left(s+b\right)^{2}}$$