A linear function f satisfies f(2)=15 and f(6)=31. What is the value of f(10)?

Because $f$ is linear, its slope is the change in output over the change in input: $\frac{31-15}{6-2} = \frac{16}{4} = 4$. Going from $x = 6$ to $x = 10$ is an increase of 4 in the input, so the output rises by $4 \cdot 4 = 16$. Therefore $f(10) = 31 + 16 = 47$. A common wrong answer is 39, which comes from adding the slope only once (31 + 4 + 4) or adding 8. Another is 35, from adding 4 to 31 once. Always multiply the slope by the full change in $x$.