For the data given​ below, answer parts ​(a) through ​(f).
x
negative 2
negative 1
0
1
2
y
negative 5
0
2
4
5

### Given Data:

| **x** | -2 | -1 |  0 | 1 | 2 |
|:-----:|----|----|----|---|---|
| **y** | -5 |  0 |  2 | 4 | 5 |

### Part (a): **Find the slope of the line passing through the points.**

To find the slope \( m \), we use the formula:

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

Using the points \( (x_1, y_1) = (-2, -5) \) and \( (x_2, y_2) = (2, 5) \):

\[
m = \frac{5 - (-5)}{2 - (-2)} = \frac{5 + 5}{2 + 2} = \frac{10}{4} = 2.5
\]

### Part (b): **Find the y-intercept of the line.**

The equation of a line is:

\[
y = mx + b
\]

We know the slope \( m = 2.5 \). To find the \( y \)-intercept \( b \), we substitute any point. Let's use \( (x, y) = (0, 2) \):

\[
2 = 2.5(0) + b \implies b = 2
\]

So, the y-intercept is \( b = 2 \).

### Part (c): **Find the equation of the line in slope-intercept form.**

The equation of the line is:

\[
y = 2.5x + 2
\]

### Part (d): **Predict the value of \( y \) when \( x = 3 \).**

Substitute \( x = 3 \) into the equation of the line:

\[
y = 2.5(3) + 2 = 7.5 + 2 = 9.5
\]

So, when \( x = 3 \), \( y = 9.5 \).

### Part (e): **Determine if the relationship between \( x \) and \( y \) is linear.**

The relationship is linear because the slope between any two points remains constant, and the data fits the equation \( y = 2.5x + 2 \), which is a linear equation.

### Part (f): **Describe the behavior of \( y \) as \( x \) increases.**

As \( x \) increases, \( y \) increases at a constant rate of 2.5 units for each 1 unit increase in \( x \). This indicates a positive, linear relationship between \( x \) and \( y \).

---

Let me know if you need more details or have any questions!

Here are 5 related questions to deepen your understanding:
1. How would the equation change if the slope was 3 instead of 2.5?
2. Can you use the slope formula to find the slope between other points, such as \( (0, 2) \) and \( (1, 4) \)?
3. What happens to the equation if the y-intercept changes to \( -1 \)?
4. How do you calculate the slope when given only two points in general?
5. Could you graph the points to visually confirm the linearity?

**Tip:** Always verify if data points align with the predicted equation by checking multiple pairs of points to confirm the slope and intercept.

Solve a math problem involving data points x = [-2, -1, 0, 1, 2] and y = [-5, 0, 2, 4, 5]. Learn how to find the slope, y-intercept, and equation of a line. This problem covers linear relationships and the slope-intercept form, suitable for Grades 8-10.

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