A ride-hailing app charges a base fee plus a cost per mile. The table shows the total charge, , in dollars, for two rides that traveled miles. | (miles) | (dollars) | |---|---| | 2 | 14 | | 7 | 29 | Assuming a linear relationship between and , which equation represents this relationship?

Solution available with step-by-step explanation

SAT Linear Functions Question 131 | Free SAT Prep | Aniko

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Question 131·Medium·Linear Functions

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A ride-hailing app charges a base fee plus a cost per mile. The table shows the total charge, $C(d)$ , in dollars, for two rides that traveled $d$ miles.

$d$ (miles)	$C(d)$ (dollars)
2	14
7	29

Assuming a linear relationship between $d$ and $C(d)$ , which equation represents this relationship?

When a problem gives a linear relationship in a table, quickly find the slope using m = (y_2 - y_1)/(x_2 - x_1), then plug one point into y = mx + b to solve for b and write the full equation. On multiple-choice SAT questions, you can also double-check by plugging the given d-values into your equation (or each answer choice) to see which one reproduces the table values exactly.

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Use the idea of base fee plus cost per mile

A base fee plus a cost per mile can be written like $C(d)=md+b$ , where $m$ is the cost per mile and $b$ is the base fee. Think about how you can find $m$ and $b$ from the table.

Find the cost per mile first

Look at how much the cost changes when the distance increases from 2 miles to 7 miles. How much extra money is charged for those extra miles? Divide the change in cost by the change in distance.

Then find the base fee

Once you know the cost per mile, plug in one of the table values (like $d=2$ , $C(d)=14$ ) into $C(d)=md+b$ to solve for $b$ , the base fee.

Check against the answer choices

After you find $m$ and $b$ , write the equation $C(d)=md+b$ using your numbers and see which answer choice has the same equation.

Desmos Guide

Use Desmos to find the cost per mile (slope)

In an expression line, type (29-14)/(7-2) and look at the value Desmos gives. This number is the cost per mile $m$ in the equation $C(d)=md+b$ .

Use Desmos to find the base fee

In a new expression line, type 14 - [value from step 1]*2 (replacing [value from step 1] with the number you saw). The result is the base fee $b$ in $C(d)=md+b$ .

Form the equation and compare to choices

Write the linear equation $C(d)=md+b$ using the $m$ and $b$ values you just found from Desmos, then match this equation to the answer choice that has the same form.

Step-by-step Explanation

Translate the situation into a linear model

For a base fee plus a cost per mile, the total cost $C(d)$ as a function of miles $d$ can be written in slope-intercept form:

C(d) = md + b

$m$ is the cost per mile (the rate of change).
$b$ is the base fee (the cost when $d=0$ ).

Find the cost per mile (the slope)

Use the two points from the table: $(2,14)$ and $(7,29)$ .

Compute the slope $m$ :

m = \frac{29 - 14}{7 - 2} = \frac{15}{5} = 3

So the ride costs $3$ dollars per mile.

Find the base fee (the y-intercept)

Use one of the points and the slope $m=3$ in $C(d)=md+b$ to solve for $b$ .

Using $(2,14)$ :

14 = 3(2) + b \\[0.7em] 14 = 6 + b \\[0.7em] b = 14 - 6 = 8

So the base fee is $8$ dollars.

Write the equation and match it to the choices

Substitute $m=3$ and $b=8$ into $C(d)=md+b$ :

C(d) = 3d + 8

Among the answer choices, this is choice D) $C(d)=3d+8$ .

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Desmos Guide

Step-by-step Explanation

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Step-by-step Explanation