To determine whether the relationship between the \( x \) and \( y \) values is linear, we can analyze the data:
\[ \begin{align*} x & : 1, 5, 9, 13 \ y & : 5, 10, 15, 20 \ \end{align*} \]
Step 1: Check the differences
Calculate the differences in \( y \) values for equal differences in \( x \) values.
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From \( x = 1 \) to \( x = 5 \):
- \( y \) changes from 5 to 10, which is an increase of \( 10 - 5 = 5 \).
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From \( x = 5 \) to \( x = 9 \):
- \( y \) changes from 10 to 15, which is an increase of \( 15 - 10 = 5 \).
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From \( x = 9 \) to \( x = 13 \):
- \( y \) changes from 15 to 20, which is an increase of \( 20 - 15 = 5 \).
Step 2: Check the ratio of change
The \( y \) values increase by 5 for each corresponding increase of 4 in \( x \) (from \( x = 1 \) to \( x = 5 \) is an increase of 4, and the same for the subsequent pairs).
Step 3: Determine if it’s linear
Since the change in \( y \) is constant when \( x \) increases by equal amounts, the relationship is linear.
Step 4: Calculate the slope
The slope \( m \) can be calculated as: \[ m = \frac{\Delta y}{\Delta x} = \frac{5}{4} = 1.25 \]
Step 5: Find the equation of the line
Using the slope-intercept form of the equation of a line \( y = mx + b \):
- We can use any point to find \( b \). Let's use the point \( (1, 5) \): \[ 5 = 1.25(1) + b \ b = 5 - 1.25 = 3.75 \]
Thus, the equation of the line is: \[ y = 1.25x + 3.75 \]
In conclusion, the relationship shown by the data is linear, and the equation modeling the data is: \[ \boxed{y = 1.25x + 3.75} \]
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