Question

Create a linear model for the data in the table.
x
4
6
8
10
12
14

y
7
16
20
28
38
43
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.)

Answer 1

To create a linear model for the given data, we need to determine the relationship between \( x \) and \( y \). The general form of a linear equation is:

\[
y = mx + b
\]

where:
- \( m \) is the slope of the line,
- \( b \) is the y-intercept.

### Step 1: Calculate the slope (\( m \))

The slope \( m \) is calculated using the formula:

\[
m = \frac{(y_2 - y_1)}{(x_2 - x_1)}
\]

Using two points from the given data, let’s pick \( (4, 7) \) and \( (14, 43) \):

\[
m = \frac{(43 - 7)}{(14 - 4)} = \frac{36}{10} = 3.6
\]

### Step 2: Calculate the y-intercept (\( b \))

Now, we'll use one of the points to solve for \( b \). Let's use the first point \( (4, 7) \):

\[
7 = 3.6(4) + b
\]

Calculating \( 3.6(4) \):

\[
3.6 \times 4 = 14.4
\]

Now substitute back to find \( b \):

\[
7 = 14.4 + b
\]
\[
b = 7 - 14.4 = -7.4
\]

### Step 3: Write the linear model

Now we can write the linear model as:

\[
y = 3.6x - 7.4
\]

### Final Answer

Thus, the linear model for the data in the table is:

\[
\boxed{y = 3.6x - 7.4}
\]