Question 625·Medium·Math

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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.

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

Which equation represents the linear relationship between $c$ and $p$ ?

A table showing a linear relationship has a constant change in output for each additional input. Put the number of cars in Desmos’s input column and the train’s capacity in its output column, then fit a line to find the rate and starting value. To match an equation with both variables on one side, rearrange carefully: moving a positive starting value changes its sign.

Hints

Hint 1
The input is the number of cars, and the output is the maximum number of people. Keep each car count paired with its capacity when you enter the table in Desmos.
Hint 2
A line’s slope tells you how much the capacity changes per additional car. The other fitted number is the capacity when the car count is $0$ ; none of the table rows gives that value directly.
Hint 3
Once you have an equation for $p$ , move terms to match the choices. Subtract from both sides rather than changing a sign by sight. What happens to the starting value when you move it?

Step-by-step

Approach 1: Fit the table, then rearrange
Step 1
Pair the cars with their capacities
Make a Desmos table. Put the number of cars in $x_1$ and the corresponding maximum number of people in $y_1$ . Keep the values from each row together so Desmos fits capacity as a function of cars.
Step 2
Find the rate and starting value
Type $y_1\sim mx_1+b$ below the table. The symbol $\sim$ tells Desmos to fit a line. Under PARAMETERS, it shows $m=55$ and $b=9$ . The slope $m$ means 55 more people per additional car; $b$ is the capacity the line gives when the car count is $0$ .
Step 3
Write the relationship using the problem’s letters
Here $c$ is the car count and $p$ is the maximum number of people, so the fitted line becomes:
$55c+9=p.$
The per-car rate multiplies $c$ , not $p$ .
Step 4
Move the starting value
Subtract $9$ from both sides:
$55c=p-9.$
The positive starting value is now negative because you subtracted it; the rate $55$ has not changed.
Step 5
Put both variables on one side
Subtract $p$ from both sides:
$55c-p=-9.$
This equation gives the train’s maximum number of passengers and crew from its number of cars. Choice D.

Approach 2: Test a table row in the choices
Step 1
Check the choices that multiply cars by 55
The first table row gives the pair $(c,p)=(3,174)$ . Type $55(3)-174$ in Desmos; it shows $-9$ . So a choice with $55c-p$ on the left must have $-9$ on the right to fit this row.
Step 2
Rule out the reversed variables
Keep the first line and type $55(174)-3$ . Desmos shows $9{,}567$ , which is neither $9$ nor $-9$ . So choices with $55p-c$ on the left fail the same row. The equation that fits is $55c-p=-9$ . Choice D.

Question 625·Medium·Math

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