Solve 4+x/4+y+5+x/y+5

$$\frac{ 4+x }{ 4+y } + \frac{ 5+x }{ y+5 }$$

Evaluate

$$\frac{2xy+9x+9y+40}{\left(y+4\right)\left(y+5\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 4+y and y+5 is \left(y+4\right)\left(y+5\right). Multiply \frac{4+x}{4+y} times \frac{y+5}{y+5}. Multiply \frac{5+x}{y+5} times \frac{y+4}{y+4}.

$$\frac{\left(4+x\right)\left(y+5\right)}{\left(y+4\right)\left(y+5\right)}+\frac{\left(5+x\right)\left(y+4\right)}{\left(y+4\right)\left(y+5\right)}$$

Since \frac{\left(4+x\right)\left(y+5\right)}{\left(y+4\right)\left(y+5\right)} and \frac{\left(5+x\right)\left(y+4\right)}{\left(y+4\right)\left(y+5\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(4+x\right)\left(y+5\right)+\left(5+x\right)\left(y+4\right)}{\left(y+4\right)\left(y+5\right)}$$

Do the multiplications in \left(4+x\right)\left(y+5\right)+\left(5+x\right)\left(y+4\right).

$$\frac{4y+20+xy+5x+5y+20+xy+4x}{\left(y+4\right)\left(y+5\right)}$$

Combine like terms in 4y+20+xy+5x+5y+20+xy+4x.

$$\frac{9y+40+2xy+9x}{\left(y+4\right)\left(y+5\right)}$$

Expand \left(y+4\right)\left(y+5\right).

$$\frac{9y+40+2xy+9x}{y^{2}+9y+20}$$

Expand

$$\frac{2xy+9x+9y+40}{\left(y+4\right)\left(y+5\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 4+y and y+5 is \left(y+4\right)\left(y+5\right). Multiply \frac{4+x}{4+y} times \frac{y+5}{y+5}. Multiply \frac{5+x}{y+5} times \frac{y+4}{y+4}.

$$\frac{\left(4+x\right)\left(y+5\right)}{\left(y+4\right)\left(y+5\right)}+\frac{\left(5+x\right)\left(y+4\right)}{\left(y+4\right)\left(y+5\right)}$$

Since \frac{\left(4+x\right)\left(y+5\right)}{\left(y+4\right)\left(y+5\right)} and \frac{\left(5+x\right)\left(y+4\right)}{\left(y+4\right)\left(y+5\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(4+x\right)\left(y+5\right)+\left(5+x\right)\left(y+4\right)}{\left(y+4\right)\left(y+5\right)}$$

Do the multiplications in \left(4+x\right)\left(y+5\right)+\left(5+x\right)\left(y+4\right).

$$\frac{4y+20+xy+5x+5y+20+xy+4x}{\left(y+4\right)\left(y+5\right)}$$

Combine like terms in 4y+20+xy+5x+5y+20+xy+4x.

$$\frac{9y+40+2xy+9x}{\left(y+4\right)\left(y+5\right)}$$

Expand \left(y+4\right)\left(y+5\right).

$$\frac{9y+40+2xy+9x}{y^{2}+9y+20}$$