SAT Question 9551cd91: Linear Functions answer and explanation with Desmos solution

Choice A is correct. It's given that there is a linear relationship between the number of cars, $c$, on a commuter train and the maximum number of passengers and crew, $p$, that the train can carry. It follows that this relationship can be represented by an equation of the form $p=mc+b$, where $m$ is the rate of change of $p$ in this relationship and $b$ is a constant. The rate of change of $p$ in this relationship can be calculated by dividing the difference in any two values of $p$ by the difference in the corresponding values of $c$. Using two pairs of values given in the table, the rate of change of $p$ in this relationship is $\frac{284-174}{5-3}$, or $55$. Substituting $55$ for $m$ in the equation $p=mc+b$ yields $p=55c+b$. The value of $b$ can be found by substituting any value of $c$ and its corresponding value of $p$ for $c$ and $p$, respectively, in this equation. Substituting $10$ for $c$ and $559$ for $p$ yields $559=55\left(10\right)+b$, or $559=550+b$. Subtracting $550$ from both sides of this equation yields $9=b$. Substituting $9$ for $b$ in the equation $p=55c+b$ yields $p=55c+9$. Subtracting $9$ from both sides of this equation yields $p-9=55c$. Subtracting $p$ from both sides of this equation yields $-9=55c-p$, or $55c-p=-9$.