## Algebra 2 Common Core

$2.5$
Recall: An inverse variation is represented by the equation $xy=k$, where $k$ is the constant of variation. As the value of $x$ increases, the value of $y$ decreases. The equation that models the inverse variation can be determined by finding the value of $k$: This can be done by taking any ordered pair $(x, y)$ then substituting the values of $x$ and $y$ into the inverse variation formula $xy=k$. Since the variation contains the point $\left(2, 5\right)$, substitute these into the formula $xy=k$ to obtain: \begin{align*} xy&=k\\\\ 2(5)&=k\\\\ 10&=k\\\\ \end{align*} Thus, the equation of the inverse variation is $xy=10$. To find the value of $y$ when $x=4$, substitute $4$ to $x$ in the equation above to obtain: \begin{align*} xy&=10\\\\ 4(y)&=10\\\\ \frac{4(y)}{4}&=\frac{10}{4}\\\\ y&=\frac{5}{2}=2.5 \end{align*}