Question
Create a linear model for the data in the table.
x
4
6
8
10
12
14
y
7
14
21
28
36
45
Question content area bottom
Part 1
Write a linear model for the data in the table.
yequals
enter your response herexplus(
enter your response here)
(Type integers or decimals rounded to three decimal places as needed.)
x
4
6
8
10
12
14
y
7
14
21
28
36
45
Question content area bottom
Part 1
Write a linear model for the data in the table.
yequals
enter your response herexplus(
enter your response here)
(Type integers or decimals rounded to three decimal places as needed.)
Answers
GPT-4o mini
To create a linear model from the data provided in the table, we need to find the linear relationship between \(x\) and \(y\). The form of a linear equation is:
\[
y = mx + b
\]
where \(m\) is the slope and \(b\) is the y-intercept.
### Step 1: Calculate the slope (m)
To find the slope, we can use the formula:
\[
m = \frac{(y_2 - y_1)}{(x_2 - x_1)}
\]
We can calculate \(m\) using any two points. Let's choose the first two points where \( (x_1, y_1) = (4, 7) \) and \( (x_2, y_2) = (6, 14) \):
\[
m = \frac{(14 - 7)}{(6 - 4)} = \frac{7}{2} = 3.5
\]
### Step 2: Find the y-intercept (b)
Next, we can use one of the points and the slope to find the y-intercept \(b\). Using the point \( (4, 7) \):
\[
y = mx + b
\]
Substituting known values:
\[
7 = 3.5(4) + b
\]
Calculating \(3.5(4)\):
\[
7 = 14 + b
\]
Rearranging to find \(b\):
\[
b = 7 - 14 = -7
\]
### Final Model
Now, we can write the linear model:
\[
y = 3.5x - 7
\]
Thus, the final answer for your linear model is:
\[
y = 3.5x + (-7)
\]
So to input this into your specified format:
\[
y = 3.5 x + (-7)
\]
Here’s a recap:
- \(m = 3.5\)
- \(b = -7\)
You should enter:
- \(m = 3.5\)
- \(b = -7\)
\[
y = mx + b
\]
where \(m\) is the slope and \(b\) is the y-intercept.
### Step 1: Calculate the slope (m)
To find the slope, we can use the formula:
\[
m = \frac{(y_2 - y_1)}{(x_2 - x_1)}
\]
We can calculate \(m\) using any two points. Let's choose the first two points where \( (x_1, y_1) = (4, 7) \) and \( (x_2, y_2) = (6, 14) \):
\[
m = \frac{(14 - 7)}{(6 - 4)} = \frac{7}{2} = 3.5
\]
### Step 2: Find the y-intercept (b)
Next, we can use one of the points and the slope to find the y-intercept \(b\). Using the point \( (4, 7) \):
\[
y = mx + b
\]
Substituting known values:
\[
7 = 3.5(4) + b
\]
Calculating \(3.5(4)\):
\[
7 = 14 + b
\]
Rearranging to find \(b\):
\[
b = 7 - 14 = -7
\]
### Final Model
Now, we can write the linear model:
\[
y = 3.5x - 7
\]
Thus, the final answer for your linear model is:
\[
y = 3.5x + (-7)
\]
So to input this into your specified format:
\[
y = 3.5 x + (-7)
\]
Here’s a recap:
- \(m = 3.5\)
- \(b = -7\)
You should enter:
- \(m = 3.5\)
- \(b = -7\)