Question

x y
1 5
5 10
9 15
13 20

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

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




The relationship is not linear?




2

Answer 1

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.

1. From \( x = 1 \) to \( x = 5 \):
- \( y \) changes from 5 to 10, which is an increase of \( 10 - 5 = 5 \).

2. From \( x = 5 \) to \( x = 9 \):
- \( y \) changes from 10 to 15, which is an increase of \( 15 - 10 = 5 \).

3. 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:
\[
\boxed{y = 1.25x + 3.75}
\]