Create a linear model for the data in the table.

Question

x
4
6
8
10
12
14

y
7
16
20
28
38
43
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Part 1
Write a linear model for the data in the table.
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​(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:

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