SAT Question 2b15d65f: Linear Functions answer and explanation with Desmos solution

Choice A is correct. Let the economist's model be the linear function $Q=mP+b$, where $Q$ is the demand, $P$ is the selling price, $m$ is the slope of the line, and $b$ is the $y$-coordinate of the $y$-intercept of the line in the $xy$-plane, where $y=Q$. Two pairs of the selling price $P$ and the demand $Q$ are given. Using the coordinate pairs $(P,Q)$, two points that satisfy the function are $(40,20,000)$ and $(60,15,000)$. The slope $m$ of the function can be found using the formula $m=\frac{Q_2-Q_1}{P_2-P_1}$. Substituting the given values into this formula yields $m=\frac{15,000-20,000}{60-40}$, or $m=-250$. Therefore, $Q=-250P+b$. The value of $b$ can be found by substituting one of the points into the function. Substituting the values of $P$ and $Q$ from the point $(40,20,000)$ yields $20,000=-250\times 40+b$, or $20,000=-10,000+b$. Adding $10,000$ to both sides of this equation yields $b=30,000$. Therefore, the linear function the economist used as the model is $Q=-250P+30,000$. Substituting $55$ for $P$ yields $Q=-250\times 55+30,000=16,250$. It follows that when the selling price is $$55$ per unit, the demand is $16,250$ units.