Question

9.
Is the relationship shown by the data linear? If so, model the data with an equation.

x y
1 5
5 10
9 15
13 20

Answer 1

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:

\[
y = 1.25x + 3.75
\]