Final answer:
To demonstrate that the points (4,2), (7,5), and (9,7) are collinear, we compare the slopes between each pair of points to confirm they are equal. The slopes are found to be equal, indicating that the points are collinear.
Step-by-step explanation:
To show that the points (4,2), (7,5), and (9,7) are collinear, we need to verify whether the slopes between any two pairs of points are equal. The slope of a line through two points (x1, y1) and (x2, y2) is calculated as (y2 - y1)/(x2 - x1).
For the points (4,2) and (7,5), the slope is (5 - 2)/(7 - 4) = 3/3 = 1. For the points (7,5) and (9,7), the slope is (7 - 5)/(9 - 7) = 2/2 = 1. Since both calculated slopes are equal, the points lie on the same line, thus, they are collinear.
Another way to confirm that the points are collinear is to use the determinant of a matrix comprising the coordinates of the points and their corresponding ones. If the determinant of this matrix is zero, then the points are collinear. The matrix for our points and the calculation of its determinant is beyond the scope of this answer, but it is an alternative method to verify collinearity.