The table below shows the remaining balance, B, in dollars, on a transit card after a user has taken r rides on a city bus. | Number of rides (r) | Remaining balance (B, in dollars) | | :---: | :---: | | 3 | 22.50 | | 8 | 10.00 | | 12 | 0.00 | Which of the following equations represents the relation

The table below shows the remaining balance, B, in dollars, on a transit card after a user… — SAT

The table below shows the remaining balance, $B$ , in dollars, on a transit card after a user has taken $r$ rides on a city bus.

Number of rides ( $r$ )	Remaining balance ( $B$ , in dollars)
$3$	$22.50$
$8$	$10.00$
$12$	$0.00$

Which of the following equations represents the relationship between $B$ and $r$ ?

A
$B = 22.50 - 2.50r$
B
$B = 30.00 - 0.40r$
$B = 30.00 - 2.50r$ Answer
D
$B = 30.00 + 2.50r$

Answer

B = 30.00 - 2.50r

The correct equation has a rate of change of

-2.50

dollars per ride and a starting balance of

30.00

dollars. The rate of change (slope) can be calculated by finding the change in balance divided by the change in the number of rides between two points, such as

(3, 22.50)

and

(8, 10.00)

, which gives

\frac{10.00 - 22.50}{8 - 3} = -2.50

. Using the slope-intercept form

B = mr + B_0

and the point

(12, 0.00)

, we get

0.00 = -2.50(12) + B_0

, which simplifies to

B_0 = 30.00

. Therefore, the equation is

B = 30.00 - 2.50r

Step-by-Step Solution

Calculate the slope (rate of change) using two points from the table, such as

(3, 22.50)

and

(8, 10.00)

The slope is

m = \frac{10.00 - 22.50}{8 - 3} = \frac{-12.50}{5} = -2.50

The slope represents the constant change in the remaining balance per ride.

Use the slope-intercept form

B = mr + B_0

and one point to find the initial balance

B_0

Substituting

m = -2.50

and the point

(12, 0.00)

gives

0.00 = -2.50(12) + B_0

, which simplifies to

0.00 = -30.00 + B_0

, so

B_0 = 30.00

The initial balance

B_0

is the y-intercept, which is the value of

B

when

r = 0

Write the final linear equation representing the relationship.

The final equation is

B = 30.00 - 2.50r

This equation models the remaining balance on the card as a function of the number of rides taken.

Key Concept

Finding a linear equation in two variables from a table of values.

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