Solve 9yv/y^2-v^2+y-v/y+v

$$\frac { 9 y v } { y ^ { 2 } - v ^ { 2 } } + \frac { y - v } { y + v }$$

Evaluate

$$\frac{y^{2}+7vy+v^{2}}{y^{2}-v^{2}}$$

Short Solution Steps

Factor y^{2}-v^{2}.

$$\frac{9yv}{\left(y+v\right)\left(y-v\right)}+\frac{y-v}{y+v}$$

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

$$\frac{9yv}{\left(y+v\right)\left(y-v\right)}+\frac{\left(y-v\right)\left(y-v\right)}{\left(y+v\right)\left(y-v\right)}$$

Since \frac{9yv}{\left(y+v\right)\left(y-v\right)} and \frac{\left(y-v\right)\left(y-v\right)}{\left(y+v\right)\left(y-v\right)} have the same denominator, add them by adding their numerators.

$$\frac{9yv+\left(y-v\right)\left(y-v\right)}{\left(y+v\right)\left(y-v\right)}$$

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

$$\frac{9yv+y^{2}-yv-yv+v^{2}}{\left(y+v\right)\left(y-v\right)}$$

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

$$\frac{v^{2}+7yv+y^{2}}{\left(y+v\right)\left(y-v\right)}$$

Expand \left(y+v\right)\left(y-v\right).

$$\frac{v^{2}+7yv+y^{2}}{y^{2}-v^{2}}$$

Expand

$$\frac{y^{2}+7vy+v^{2}}{y^{2}-v^{2}}$$

Short Solution Steps

Factor y^{2}-v^{2}.

$$\frac{9yv}{\left(y+v\right)\left(y-v\right)}+\frac{y-v}{y+v}$$

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

$$\frac{9yv}{\left(y+v\right)\left(y-v\right)}+\frac{\left(y-v\right)\left(y-v\right)}{\left(y+v\right)\left(y-v\right)}$$

Since \frac{9yv}{\left(y+v\right)\left(y-v\right)} and \frac{\left(y-v\right)\left(y-v\right)}{\left(y+v\right)\left(y-v\right)} have the same denominator, add them by adding their numerators.

$$\frac{9yv+\left(y-v\right)\left(y-v\right)}{\left(y+v\right)\left(y-v\right)}$$

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

$$\frac{9yv+y^{2}-yv-yv+v^{2}}{\left(y+v\right)\left(y-v\right)}$$

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

$$\frac{v^{2}+7yv+y^{2}}{\left(y+v\right)\left(y-v\right)}$$

Expand \left(y+v\right)\left(y-v\right).

$$\frac{v^{2}+7yv+y^{2}}{y^{2}-v^{2}}$$