Find an equation of the line of best fit for the data. Round the slope and the $y$ -intercept to the nearest tenth, if necessary.
$x$ 7 8 10 13 15
$y$ 25 29 41 48 57
To find the equation of the line of best fit, we first need to determine the slope and the $y$-intercept.
We can find the slope using the formula $m = \dfrac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2}$, where $n$ is the number of data points and the sums are over all the data points. Plugging in the given values, we have \begin{align*}
n &= 5, \\
\sum x &= 7+8+10+13+15 = 53, \\
\sum y &= 25+29+41+48+57 = 200, \\
\sum x^2 &= (7)^2 + (8)^2 + (10)^2 + (13)^2 + (15)^2 = 583, \\
\sum xy &= (7)(25)+(8)(29)+(10)(41)+(13)(48)+(15)(57) = 2466.
\end{align*} Therefore, we have \begin{align*}
m &= \dfrac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2} \\
&= \dfrac{(5)(2466)-(53)(200)}{(5)(583)-(53)^2} \\
&= \dfrac{12330-10600}{2915-2809} \\
&= \dfrac{1730}{106} \\
&= 16.3.
\end{align*}
Now we need the $y$-intercept. Since the line of best fit goes through the point $(\overline{x},\overline{y})$, where $\overline{x}=\dfrac{\sum x}{n}$ and $\overline{y}=\dfrac{\sum y}{n}$, and we just computed $\sum x=53$ and $\sum y=200$, we have $\overline{x} = \dfrac{53}{5} = 10.6$ and $\overline{y} = \dfrac{200}{5} = 40$.
Therefore the line of best fit has slope $m=16.3$ and $y$-intercept $(\overline{x},\overline{y})=(10.6,40)$.
Therefore our equation is $y = 16.3(x-10.6)+40 = 16.3x-172.78+40 = 16.3x-132.78$, or $\boxed{y = 16.3x-132.78}$.