Answer
$y=\dfrac{225}{7}x+\dfrac{12,350}{7}$
Work Step by Step
We have to determine the equation
$$y=mx+b,$$
where
$y$ represents the artifact's age
$x$ represents the artifact's depth.
We are given two points on the graph of the line describing the equation: $(54,3500)$ and $(26,2600)$.
$\textbf{First method}$
We will write the equation in point-slope form, then rewrite it in slope-intercept form.
We calculate the slope:
$$m=\dfrac{y_2-y_1}{x_2-x_1}=\dfrac{3500-2600}{54-26}=\dfrac{225}{7}.$$
We determine the point-slope equation using the slope $m$ and one of the points, $(26,2600)$:
$$y-y_0=m(x-x_0)$$
$$y-2600=\dfrac{225}{7}(x-26)$$
Rewrite the equation in slope-intercept form:
$$y=\dfrac{225}{7}x-\dfrac{5850}{7}+2600$$
$$y=\dfrac{225}{7}x+\dfrac{12,350}{7}.$$
$\textbf{Second method}$
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:
$$m=\dfrac{y_2-y_1}{x_2-x_1}=\dfrac{3500-2600}{54-26}=\dfrac{225}{7}.$$
Substitute the slope and the coordinates of one point, for example $(26,2600)$, into the slope-intercept form and solve for $b$:
$$\begin{align*}
y&=mx+b\\
2600&=\dfrac{225}{7}(26)+b\\
b&=2600-\dfrac{5850}{7}=\dfrac{12,350}{7}.
\end{align*}$$
Substitute $m$ and $b$ into the slope-intercept form:
$$y=\dfrac{225}{7}x+\dfrac{12,350}{7}.$$
