Is this linear?  X: 0,1,2,3  Y: 4,12,20,28  

To determine whether the data given is linear, we need to check if there is a constant rate of change between the \(X\) and \(Y\) values, which is equivalent to checking if the slope between consecutive points is the same.

You have the following pairs of \((X, Y)\) values:
- \( (0, 4) \)
- \( (1, 12) \)
- \( (2, 20) \)
- \( (3, 28) \)

### Step 1: Calculate the slope between consecutive points

The slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

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

Let's calculate the slope between each consecutive pair of points:

1. **Between \( (0, 4) \) and \( (1, 12) \):**

\[
m_1 = \frac{12 - 4}{1 - 0} = \frac{8}{1} = 8
\]

2. **Between \( (1, 12) \) and \( (2, 20) \):**

\[
m_2 = \frac{20 - 12}{2 - 1} = \frac{8}{1} = 8
\]

3. **Between \( (2, 20) \) and \( (3, 28) \):**

\[
m_3 = \frac{28 - 20}{3 - 2} = \frac{8}{1} = 8
\]

### Step 2: Conclusion
Since the slope is constant (\(m = 8\)) between all consecutive points, the relationship between \(X\) and \(Y\) is **linear**.

### Equation of the Line
Now, we can find the equation of the line. Since the slope \(m = 8\) and we have a point \( (0, 4) \), we can use the slope-intercept form \(y = mx + c\), where \(c\) is the y-intercept.

Substitute the known values:

\[
y = 8x + 4
\]

So, the equation of the line is:

\[
y = 8x + 4
\]

Would you like to go over any details or have further questions?

Here are five related questions:
1. How do you find the slope between two points?
2. What are the characteristics of a

Find out whether the relationship between X: 0,1,2,3 and Y: 4,12,20,28 is linear by calculating the slope between consecutive points. This problem explores key algebraic concepts such as slope calculation and the linear equation. Suitable for students in Grades 6-8.

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