Answer
$y=\dfrac{45}{13}x+\dfrac{60}{13}$
Work Step by Step
We have to determine the equation
$$y=mx+b,$$
where
$y$ represents the number of calories burnt per hour
$x$ represents the swimmer's weight.
We are given two points on the graph of the line describing the equation: $(120,420)$ and $(172,600)$.
$\textbf{First method}$
We will write the equation in point-slope form, then rewrite it in slope-intercept form.
We calculate the slope:
$$m=\dfrac{y_2-y_1}{x_2-x_1}=\dfrac{600-420}{172-120}=\dfrac{45}{13}.$$
We determine the point-slope equation using the slope $m$ and one of the points, $(120,420)$:
$$y-y_0=m(x-x_0)$$
$$y-420=\dfrac{45}{13}(x-120)$$
Rewrite the equation in slope-intercept form:
$$y=\dfrac{45}{13}x+\dfrac{60}{13}.$$
$\textbf{Second method}$
We will calculate the slope of the line, then its $y$-intercept and finally write the equation in slope-intercept form.
We calculate the slope:
$$m=\dfrac{y_2-y_1}{x_2-x_1}=\dfrac{600-420}{172-120}=\dfrac{45}{13}.$$
Substitute the slope and the coordinates of one point, for example $(120,420)$, into the slope-intercept form and solve for $b$:
$$\begin{align*}
y&=mx+b\\
420&=\dfrac{45}{13}(120)+b\\
b&=420-\dfrac{5400}{13}=\dfrac{60}{13}.
\end{align*}$$
Substitute $m$ and $b$ into the slope-intercept form:
$$y=\dfrac{45}{13}x+\dfrac{60}{13}.$$
