Context: The table gives three points of a linear relationship: $(0, n)$, $(4, n + 19)$, and $(8, n + 38)$, where $n$ is a constant.

Answer: $\dfrac{19}{4}$

Why: The slope of a line is $\dfrac{\text{change in } y}{\text{change in } x}$ between any two points. From $x = 0$ to $x = 4$, $y$ goes from $n$ to $n + 19$, a change of $(n + 19) - n = 19$ while $x$ changes by $4$. So slope $= \dfrac{19}{4}$. The constant $n$ cancels, so its value is never needed.

Verification: From $x = 4$ to $x = 8$: $\dfrac{(n + 38) - (n + 19)}{8 - 4} = \dfrac{19}{4}$. The same value on both intervals confirms the relationship is linear with slope $\dfrac{19}{4}$.

Common slips:

• $19$ comes from using the $y$-change alone and forgetting that $x$ steps by $4$, not $1$.
• $\dfrac{4}{19}$ comes from inverting rise over run.