The points (1, 2), (3, 8), and (x, y) are collinear. If the third point lies on the line y = 2x - 1, find the value of x.
Correct: B
First, find the equation of the line passing through the points (1, 2) and (3, 8).
Step 1: Calculate the slope (m) of the line.
m = (y2 - y1) / (x2 - x1)
m = (8 - 2) / (3 - 1)
m = 6 / 2
m = 3.
Step 2: Use the point-slope form (y - y1 = m(x - x1)) to find the equation of the line.
Using point (1, 2) and slope m = 3:
y - 2 = 3(x - 1)
y - 2 = 3x - 3
y = 3x - 1.
Step 3: The third point (x, y) is collinear with the first two points, so it must lie on the line y = 3x - 1.
We are also given that this third point (x, y) lies on the line y = 2x - 1.
Step 4: Find the x-coordinate of the third point by setting the two y-equations equal to each other.
3x - 1 = 2x - 1
Add 1 to both sides:
3x = 2x
Subtract 2x from both sides:
x = 0.
To verify, if x=0, then y = 2(0) - 1 = -1. So the third point is (0, -1). Let's check if it lies on y = 3x - 1: y = 3(0) - 1 = -1. It does.
Thus, the value of x is 0.