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Algebra / Linear functions Difficulty: Medium

Number of cars	Maximum number of passengers and crew
$3$ $3$	$174$ $174$
$5$ $5$	$284$ $284$
$10$ $10$	$559$ $559$

The table shows the 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. Which equation represents the linear relationship between $c$ and $p$ ?

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Explanation

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 $p = m c + b$ , 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 $p = m c + b$ yields $p = 55 c + b$ . 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 $559 = 55 (10) + b$ , or $559 = 550 + b$ . Subtracting $550$ from both sides of this equation yields $9 equals b$ . Substituting $9$ for $b$ in the equation $p = 55 c + b$ yields $p = 55 c + 9$ . Subtracting $9$ from both sides of this equation yields $p - 9 = 55 c$ . Subtracting $p$ from both sides of this equation yields $- 9 = 55 c - p$ , or $55 c - p = - 9$ .

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.