Sure, let's determine the linear relationship from the given data to represent it graphically. Let's go through the process step-by-step:
### Step 1: Define the Variables
Let:
- [tex]\( x \)[/tex] be the number of months.
- [tex]\( y \)[/tex] be the total cost paid.
### Step 2: Set Up the Equations
From the table, we have the following data points:
1. [tex]\((3, 66)\)[/tex]
2. [tex]\((5, 90)\)[/tex]
3. [tex]\((8, 126)\)[/tex]
The general form of a linear equation is:
[tex]\[ y = mx + f \][/tex]
where [tex]\( m \)[/tex] is the slope and [tex]\( f \)[/tex] is the y-intercept.
### Step 3: Create a System of Equations
Using the data points, we set up the following system of equations:
1. For [tex]\( (3, 66) \)[/tex]:
[tex]\[ 66 = 3m + f \][/tex]
2. For [tex]\( (5, 90) \)[/tex]:
[tex]\[ 90 = 5m + f \][/tex]
3. For [tex]\( (8, 126) \)[/tex]:
[tex]\[ 126 = 8m + f \][/tex]
### Step 4: Solve the System of Equations
By solving the first two equations, we get:
1. [tex]\( 66 = 3m + f \)[/tex]
2. [tex]\( 90 = 5m + f \)[/tex]
Subtract Equation 1 from Equation 2:
[tex]\[ (5m + f) - (3m + f) = 90 - 66 \][/tex]
[tex]\[ 2m = 24 \][/tex]
[tex]\[ m = 12 \][/tex]
Substitute [tex]\( m = 12 \)[/tex] back into Equation 1:
[tex]\[ 66 = 3(12) + f \][/tex]
[tex]\[ 66 = 36 + f \][/tex]
[tex]\[ f = 30 \][/tex]
Therefore, the slope [tex]\( m \)[/tex] is [tex]\( 12 \)[/tex] and the y-intercept [tex]\( f \)[/tex] is [tex]\( 30 \)[/tex].
### Step 5: Write the Linear Equation
The linear equation representing the relationship is:
[tex]\[ y = 12x + 30 \][/tex]
### Step 6: Graph the Equation
To graph the equation:
1. Start at the y-intercept, [tex]\( f = 30 \)[/tex]. This point is [tex]\((0, 30)\)[/tex].
2. Use the slope [tex]\( m = 12 \)[/tex], which means for every 1 month (1 unit increase in [tex]\( x \)[/tex]), the total cost increases by 12 dollars (12 units increase in [tex]\( y \)[/tex]).
Let’s plot a few more points using the equation:
- When [tex]\( x = 3 \)[/tex]:
[tex]\[ y = 12(3) + 30 = 66 \][/tex]
- When [tex]\( x = 5 \)[/tex]:
[tex]\[ y = 12(5) + 30 = 90 \][/tex]
- When [tex]\( x = 8 \)[/tex]:
[tex]\[ y = 12(8) + 30 = 126 \][/tex]
These points, (3, 66), (5, 90), and (8, 126), should lie on the line.
The correct graph is the one that passes through these points and has a y-intercept at [tex]\( (0, 30) \)[/tex].
