Question

Q.7

Hector is considering joining a video game service in which you can rent different video games. Hector has two options:
Option 1: requires a monthly fee of $7.50 and then $3.50 per game rental fee.
Option 2: provides the following chart to show rental fees
Number of Games Rental Fee
3 $16.50
6 $33.00
9 $49.50
12 $66.00
If x is the number of games and y is the rental fee, write an equation for both of the relationships in the form .

Option 1: y = ____x +___

Option 2: y = ____x +___



If Hector wants to rent 5 video games this month, use substitution to find which option would be the least expensive, Option 1 or Option 2.

Answer 1

Option 1: y = 3.5x + 7.5

Option 2: y = 5.5x

For 5 games, Option 1 is less expensive.

Explanation:

The equation of a line in the slope-intercept form is

y = mx + b

where m = slope, and b = y-intercept.

The slope is a rate. The y-intercept is the initial value.

If a store has a fixed fee, that fixed fee is the initial value. The amount per rental is a rate, so it is the slope.

Option 1:

Monthly fee: $7.50; this is the y-intercept, b = 7.5

Per game cost: $3.50; this is a rate, $3.50/game rented; this is the slope; m = 3.5

The equation is

y = mx + b

y = 3.5x + 7.5

Option 2:

Since we are not given values for m and b directly, we calculate them. Each line in the table is an ordered pair. We can use any two ordered pairs to find the equation of a line.

Let's use these two lines in the table:

3 $16.50 as an ordered pair: (3, 16.5)

6 $33.00 as an ordered pair: (6, 33)

slope = m = (difference in y)/(difference in x)

slope = m = (33 - 16.5)/(6 - 3) = 16.5/3 = 5.5

The equation is

y = 5.5x + b

Now we use the ordered pair (3, 16.5) to find b, the y-intercept.

16.5 = 5.5(3) + b

16.5 = 16.5 + b

b = 0

Since b = 0, the equation is

y = 5.5x + 0, or simply

y = 5.5x

Now we use the two equations we found to find the cost of renting 5 video games.

Option 1:

y = 3.5x + 7.5

For 5 games, x = 5.

y = 3.5 × 5 + 7.5

y = 17.5 + 7.5

y = 25

Option 2:

y = 5.5x

For 5 games, x = 5.

y = 5.5 × 5

y = 27.5

For 5 games, Option 1 costs $25, and Option 2 costs $27.50. For 5 games, Option 1 is less expensive.