Question

By your cell phone contract, you pay a monthly fee plus some money for each minute you use the phone during the month. In one month, you spent 290 minutes on the phone, and paid $22.25. In another month, you spent 360 minutes on the phone, and paid $24.00.

Let x be the number of minutes you talk over the phone in a month, and let y be your cell phone bill for that month. Use a linear equation to model your monthly bill based on the number of minutes you talk over the phone.
a. This linear model's slope-intercept equation is_____________.
b. If you spent 140 minutes over the phone in a month, you would pay________________ .
c. If in a month, you paid $26.25 of cell phone bill, you must have spent ______________minutes on the phone in that month.

Answer 1

The linear model's slope-intercept equation is y = $0.025x + $15.00. For 140 minutes phone usage, the bill would be $18.50, and a bill of $26.25 corresponds to 450 minutes of phone usage.

Step-by-step explanation:

To answer the student's question, we'll first write two linear equations using the provided data and then solve for both the monthly fee and the cost per minute.

Step 1: Setting up the equations

For the month with 290 minutes and a bill of $22.25:
y = mx + b$22.25 = m(290) + b

For the month with 360 minutes and a bill of $24.00:
y = mx + b$24.00 = m(360) + b

Step 2: Solving the system of equations

Subtract the equations to find the cost per minute (m):
$24.00 - $22.25 = m(360) - m(290)$1.75 = 70mm = $0.025

Plug in the value of m to one of the equations to find the monthly fee (b):
$22.25 = $0.025(290) + bb = $15.00

Step 3: Final equation

The slope-intercept equation modeling the monthly bill is:
y = $0.025x + $15.00

Answer to Parts a, b, and c

• 
  
• a. The linear model's slope-intercept equation is y = $0.025x + $15.00.
• 
  
• b. If you spent 140 minutes over the phone in a month, you would pay $18.50.
• 
  
• c. If in a month, you paid $26.25 of cell phone bill, you must have spent 450 minutes on the phone in that month.

Answer 2

(a)
`y = 0.025x+15`

(b) 18.5 minutes

(c) 450.4 minutes

Step-by-step explanation:

Given

`290\ minutes = \$22.25`

`360\ minutes = \$24.00`

Solving (a): Determine the linear equation

First, we need to calculate the slope (m)

`m = (y_2 - y_1)/(x_2 - x_1)`

Where

`(x_1,y_1) = (290,22.25)`

`(x_2,y_2) = (360,24.00)`

So, we have:

`m = (24.00 - 22.25)/(360 - 290)`

`m = (1.75)/(70)`

Multiply through by 100

`m = (1.75 * 100)/(70 * 100)`

`m = (175)/(7000)`

Next, is to calculate the equation of the line using:

`y - y_1 = m(x - x_1)`

Recall that:

`(x_1,y_1) = (290,22.25)`

`m = (175)/(7000)`

`y - 22.25 = (175)/(7000)(x - 290)`

`y - 22.25 = (175x)/(7000) - (175)/(7000) * 290`

`y - 22.25 = (175x)/(7000) - (50750)/(7000)`

Add 22.25 to both sides

`y = (175x)/(7000) - (5075)/(700) + 22.25`

`y = (175x)/(7000) + (-5075 + 15575)/(700)`

`y = (175x)/(7000) + (10500)/(700)`

`y = 0.025x+15`

(b) Solve for y when x = 140

`y = 0.025x+15`

Substitute 140 for x

`y = 0.025 * 140 + 15`

`y = 3.5 + 15`

`y = 18.5`

(c) Solve for x when y = 26.25

`y = 0.025x+15`

Substitute 26.25 for y

`26.25 = 0.025x + 15`

Solve for 0.025x

`0.025x = 26.26 - 15`

`0.025x = 11.26`

Solve for x

`x = 11.26/0.025`

`x = 450.4\ minutes`