Answer:
To show that the points (7, -2), (5, 1), and (3, 4) are collinear, we need to verify that they lie on the same straight line.
We can use the slope formula to calculate the slope of the line that passes through (7, -2) and (5, 1):
m = (y2 - y1) / (x2 - x1)
m = (1 - (-2)) / (5 - 7) = -3/2
Now we can use the point-slope form of a line to write the equation of the line that passes through (7, -2) and has slope -3/2:
y - (-2) = (-3/2)(x - 7)
Simplifying this equation, we get:
y + 2 = (-3/2)x + 21/2
y = (-3/2)x + 17/2
Next, we can verify if the third point (3, 4) lies on this line. We can substitute x = 3 and y = 4 into the equation of the line:
4 = (-3/2)(3) + 17/2
4 = -9/2 + 17/2
4 = 8/2
Since this equation is true, the point (3, 4) lies on the line that passes through (7, -2) and (5, 1). Therefore, the points (7, -2), (5, 1), and (3, 4) are collinear.