Final answer:
To find the equation of a line passing through two given points, use the slope-intercept form. The equation of the line passing through points P(5,1) and Q(1,-1) is y = (1/2)x - 3/2. To show that points P, Q, and R(11,4) are collinear, substitute the coordinates of R into the equation and verify that it is satisfied.
Step-by-step explanation:
To find the equation of a line passing through two given points, we can use the slope-intercept form of the equation: y = mx + b.
Step 1: Find the slope, m, by using the formula (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are the coordinates of the two points.
Step 2: Substitute one of the points into the equation and solve for the y-intercept, b.
Step 3: Write the equation using the slope and y-intercept.
For the given points P(5,1) and Q(1,-1):
Step 1: m = (-1 - 1) / (1 - 5) = -2 / -4 = 1/2
Step 2: Using point P(5,1), substituting x = 5, y = 1 into the equation: 1 = (1/2)(5) + b. Solving for b: b = 1 - 5/2 = -3/2.
Step 3: Plugging in the values we found, the equation of the line passing through P and Q is: y = (1/2)x - 3/2.
To show that points P, Q, and R(11,4) are collinear, we can check if they lie on the same line. By substituting the x and y coordinates of point R into the equation y = (1/2)x - 3/2, we can see that the equation holds true. Therefore, P, Q, and R are collinear.