Create a linear model for the data in the table.

Question

x
5
7
9
11
13
15

y
7
16
20
28
38
45
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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 data in the table, we need to find the slope (m) and the y-intercept (b) of the line in the form \( y = mx + b \).

Given Data:

x:  5,  7,  9, 11, 13, 15
y:  7, 16, 20, 28, 38, 45

Step 1: Calculate the slope (m)

The formula for the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:

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

We can calculate the slope using any two points. I'll use the points (5, 7) and (15, 45).

\[ m = \frac{45 - 7}{15 - 5} = \frac{38}{10} = 3.8 \]

Step 2: Calculate the y-intercept (b)

To find the y-intercept, we can use the slope-intercept form \(y = mx + b\). We can rearrange it to find \(b\):

\[ b = y - mx \]

Using the point (5, 7):

\[ b = 7 - 3.8 \cdot 5 \] \[ b = 7 - 19 = -12 \]

Step 3: Write the linear model

Now we have the slope (m) and the y-intercept (b):

\[ m = 3.8, \quad b = -12 \]

So the linear model is:

\[ y = 3.8x - 12 \]

Final Answer:

\[ y = 3.8x - 12 \]