# How do you write the mixed expression t+(v+w)/(v-w) as a rational expression?

May 11, 2018

$= \frac{t \left(v - w\right) + \left(v + w\right)}{v - w}$

or

$= \frac{\left(1 + t\right) v + \left(1 - t\right) w}{v - w}$

#### Explanation:

The key here is to obtain a common denominator. This can be done by multiplying the $t$ by a factor of $1$ so that the expression does not change. But we can write our $1$ in a way that is useful.

$t + \frac{v + w}{v - w}$

$= t \cdot \frac{v - w}{v - w} + \frac{v + w}{v - w}$

$= \frac{t \left(v - w\right)}{v - w} + \frac{v + w}{v - w}$

$= \frac{t \left(v - w\right) + \left(v + w\right)}{v - w}$

If you wanted to make sure $u$ and $v$ were only written once in the numerator, you could rearrange.

$= \frac{t v - t w + v + w}{v - w}$

$= \frac{\left(1 + t\right) v + \left(1 - t\right) w}{v - w}$

Anything over a single denominator is considered a ration expression.