To determine if the relationship shown by the data is linear, we can plot the points given or look for a pattern in the (x, y) coordinates. The given data points are:
| x | y | |---|----| | 1 | 5 | | 5 | 10 | | 9 | 15 | | 13| 20 |
Step 1: Check for Linear Relationship
A relationship is linear if it can be represented by the equation of a line in the form of \( y = mx + b \), where:
- \( m \) is the slope,
- \( b \) is the y-intercept.
Step 2: Calculate the Slope \( m \)
The slope can be calculated as:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Using the first two points (1, 5) and (5, 10):
\[ m = \frac{10 - 5}{5 - 1} = \frac{5}{4} = 1.25 \]
Step 3: Determine the Y-Intercept \( b \)
Now, we can use one of the points to find \( b \). Using the point (1, 5):
\[ y = mx + b \ 5 = 1.25(1) + b \ 5 = 1.25 + b \ b = 5 - 1.25 = 3.75 \]
Step 4: Write the Equation
Thus, the relationship can be modeled with the equation:
\[ y = 1.25x + 3.75 \]
Step 5: Check Other Points
To confirm the linear model, we can check the other points:
-
For \( x = 5 \): \[ y = 1.25(5) + 3.75 = 6.25 + 3.75 = 10 \quad \text{(correct)} \]
-
For \( x = 9 \): \[ y = 1.25(9) + 3.75 = 11.25 + 3.75 = 15 \quad \text{(correct)} \]
-
For \( x = 13 \): \[ y = 1.25(13) + 3.75 = 16.25 + 3.75 = 20 \quad \text{(correct)} \]
Conclusion
The relationship is indeed linear, and it can be modeled by the equation:
\[ y = 1.25x + 3.75 \]