Answer :
To solve this problem, we need to determine the relationship between the number of text messages sent and the total cost. The equation for this relationship will be in the form of a linear equation [tex]\( y = mx + b \)[/tex], where [tex]\( y \)[/tex] is the total cost, [tex]\( x \)[/tex] is the number of text messages sent, [tex]\( m \)[/tex] is the rate of change (cost per text message), and [tex]\( b \)[/tex] is the initial value (the flat access fee).
### Step-by-Step Solution:
1. Determine the rate of change ([tex]\( m \)[/tex]):
- The rate of change (or slope [tex]\( m \)[/tex]) represents the additional cost per text message.
- To find this, we can use the values from the table.
- Let's take two points from the table: (10, 6.50) and (15, 6.75).
- The formula for the slope [tex]\( m \)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
- Plugging in our values:
[tex]\[ m = \frac{6.75 - 6.50}{15 - 10} = \frac{0.25}{5} = 0.05 \][/tex]
- Thus, the rate of change [tex]\( m \)[/tex] is 0.05, meaning the cost per text message is [tex]$0.05. 2. Determine the initial value (\( b \)): - The initial value \( b \) represents the flat access fee. - Using the formula for a line \( y = mx + b \), we will solve for \( b \). - We already know one point: (10, 6.50), and the slope \( m = 0.05 \). - Plugging these into the equation gives: \[ 6.50 = 0.05 \times 10 + b \] - Simplify and solve for \( b \): \[ 6.50 = 0.50 + b \] \[ b = 6.50 - 0.50 \] \[ b = 6.00 \] - Thus, the initial value \( b \) is 6.00, meaning the flat access fee is $[/tex]6.00.
3. Write the equation:
- Now we have [tex]\( m = 0.05 \)[/tex] and [tex]\( b = 6.00 \)[/tex].
- Therefore, the equation that represents the total cost [tex]\( y \)[/tex] for [tex]\( x \)[/tex] text messages sent is:
[tex]\[ y = 0.05x + 6.00 \][/tex]
### Explanation of the Rate of Change and Initial Value:
- Rate of Change ([tex]\( m = 0.05 \)[/tex]):
- This represents the cost per text message. For every additional text message sent, the total cost increases by [tex]$0.05. - Initial Value (\( b = 6.00 \)): - This represents the flat access fee charged by the wireless company. Even if no text messages are sent, the user has to pay a flat fee of $[/tex]6.00.
Therefore, the equation [tex]\( y = 0.05x + 6.00 \)[/tex] accurately describes the relationship between the number of text messages sent and the total cost, with [tex]\( 0.05 \)[/tex] being the cost per text message and [tex]\( 6.00 \)[/tex] being the flat access fee.
### Step-by-Step Solution:
1. Determine the rate of change ([tex]\( m \)[/tex]):
- The rate of change (or slope [tex]\( m \)[/tex]) represents the additional cost per text message.
- To find this, we can use the values from the table.
- Let's take two points from the table: (10, 6.50) and (15, 6.75).
- The formula for the slope [tex]\( m \)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
- Plugging in our values:
[tex]\[ m = \frac{6.75 - 6.50}{15 - 10} = \frac{0.25}{5} = 0.05 \][/tex]
- Thus, the rate of change [tex]\( m \)[/tex] is 0.05, meaning the cost per text message is [tex]$0.05. 2. Determine the initial value (\( b \)): - The initial value \( b \) represents the flat access fee. - Using the formula for a line \( y = mx + b \), we will solve for \( b \). - We already know one point: (10, 6.50), and the slope \( m = 0.05 \). - Plugging these into the equation gives: \[ 6.50 = 0.05 \times 10 + b \] - Simplify and solve for \( b \): \[ 6.50 = 0.50 + b \] \[ b = 6.50 - 0.50 \] \[ b = 6.00 \] - Thus, the initial value \( b \) is 6.00, meaning the flat access fee is $[/tex]6.00.
3. Write the equation:
- Now we have [tex]\( m = 0.05 \)[/tex] and [tex]\( b = 6.00 \)[/tex].
- Therefore, the equation that represents the total cost [tex]\( y \)[/tex] for [tex]\( x \)[/tex] text messages sent is:
[tex]\[ y = 0.05x + 6.00 \][/tex]
### Explanation of the Rate of Change and Initial Value:
- Rate of Change ([tex]\( m = 0.05 \)[/tex]):
- This represents the cost per text message. For every additional text message sent, the total cost increases by [tex]$0.05. - Initial Value (\( b = 6.00 \)): - This represents the flat access fee charged by the wireless company. Even if no text messages are sent, the user has to pay a flat fee of $[/tex]6.00.
Therefore, the equation [tex]\( y = 0.05x + 6.00 \)[/tex] accurately describes the relationship between the number of text messages sent and the total cost, with [tex]\( 0.05 \)[/tex] being the cost per text message and [tex]\( 6.00 \)[/tex] being the flat access fee.
