Once you know how much the price goes up when the side length increases by 3 inches, divide that change in price by 3 to find the increase in price for each 1-inch increase in side length. This is the slope.

Write an equation in the form $P = ms + b$ using the slope you found. Then plug in the $s$ and $P$ values from any one row of the table to solve for $b$.

After you find the slope and the intercept $b$, look for the choice that has both that slope and that intercept. You can also plug in one of the table's $(s, P)$ pairs to see which equation is true.

Create a table in Desmos with $s$ as $x$ and $P$ as $y$. In the first column, enter $3$, $6$, and $9$. In the second column, enter $8$, $18$, and $28$. You should see three plotted points: $(3,8)$, $(6,18)$, and $(9,28)$.

In separate expression lines, enter each option using $x$ for $s$ and $y$ for $P$, like $y = 3x + 10$, $y = \frac{10}{3}x + 8$, $y = \frac{10}{3}x - 2$, and $y = \frac{3}{10}x - \frac{1}{10}$. Desmos will draw four lines.

Look at which line passes exactly through all three plotted points $(3,8)$, $(6,18)$, and $(9,28)$. The equation of that line corresponds to the correct relationship between $s$ and $P$.

Look at how $P$ changes when $s$ increases:

• From $s=3$ to $s=6$, $P$ goes from $8$ to $18$ (an increase of $10$).
• From $s=6$ to $s=9$, $P$ goes from $18$ to $28$ (another increase of $10$).

Each time $s$ increases by $3$, $P$ increases by $10$. The slope (rate of change) of a linear function is

For a linear relationship between $P$ and $s$, we can write an equation in slope-intercept form:

where $m$ is the slope and $b$ is the $P$-intercept (the value of $P$ when $s=0$).

From Step 1, we found $m = \tfrac{10}{3}$, so the equation must look like

Pick any point from the table, for example $(s, P) = (3, 8)$, and substitute it into

Substitute $s = 3$ and $P = 8$:

Compute $\tfrac{10}{3} \cdot 3 = 10$, so

Solve for $b$:

So the $P$-intercept is $b = -2$. Any correct equation must have slope $\tfrac{10}{3}$ and intercept $-2$.

Now plug the slope and intercept into the linear form $P = ms + b$:

Comparing with the answer choices, this matches choice C, so the correct relationship between side length $s$ and price $P$ is $P = \dfrac{10}{3}s - 2$.