Final answer:
To check if the points A (1,3), B (2,6), and C (-10,-30) are collinear, we calculate the slopes AB and AC, both of which are equal to 3. This confirms that the points are collinear since they share the same slope, meaning they lie on the same straight line graph.
Step-by-step explanation:
Using Slopes to Determine Collinearity
To determine if the points A (1,3), B (2,6), and C (-10,-30) are collinear, we must calculate the slope between each pair of points and check for consistency. The slope of a line is the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line.
Let's find the slope between points A and B:
Slope of AB = (Y2 - Y1) / (X2 - X1) = (6 - 3) / (2 - 1) = 3 / 1 = 3
Now, let's calculate the slope between points A and C:
Slope of AC = (Y3 - Y1) / (X3 - X1) = (-30 - 3) / (-10 - 1) = (-33) / (-11) = 3
Since the slopes AB and AC are equal, we can confirm that points A, B, and C are indeed collinear. According to the algebra of straight lines, if the slopes between each pair of points are equal, the points lie on the same straight line. Thus, A, B, and C all share the same line graph.