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 $y$.

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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.